Linear Algebra T/F

The transpose of a sum of two matrices is the sum of the transposed matrices.

(A+B)^T=A^T+B^T. True

The matrix vector product of a 2x3 matrix and a 3xI vector is a 3 x I vector.

FALSE

The product of a matrix and a standard vector equals a standard vector.

FALSE

If the reduced row echelon form of [A b] contains a zero row, then Ax = b must have infinitely many solutions.

False it is possible if the system contains more equations as variables that the system contains one solution. It is also possible that the system has no solutions if the reduced row echelon form also contains a row with all zeros except for the last column.

If the reduced row echelon form of the augmented matrix of a system of linear equations contains a zero row, then the system is consistent.

False, A system is consistent if there is no row in this matrix in which the only nonzero entry lies in the last column; the existence of a zero row is thus no guarantee of a consistent system.

If the reduced row echelon form of [A b] contains a zero row, then Ax = b must be consistent.

False, a system is consistent if the reduced row echelon form of the augmented matrix contains a row with all zeros except for the last column. The reduced row echelon form containing a zero row is thus not a sufficient condition.

1.3 Every system of linear equations has at least one solution.

False, a system of linear equations can also have zero solutions. For example the system x+y=1 and x+y=2. This system cannot have a solution because the sum of the same two variables is both 1 and 2, which is not possible.

Some systems of linear equations have exactly two solutions.

False, a system of linear equations either have no solutions or one solution or infinitely many solutions.

If A is an mxn matrix, and u and v are vectors in R^n such that Au=Av, then u= v.

False, for example A=[1 0; 0 0], u=[1; 1] and v=[1;9] we then have that Au=[1;0]=Av, but u is not equal to v.

If A is an mxn matrix, then the only vector u in R^n such that Au= 0 is u= 0.

False, for example A=[2 -2; 2 -2] and u=[1;1], their product is then Au= [2-2;2-2]=[0;0] while the vector u is not 0.

Every vector in R^2 is a linear combination of two parallel vectors.

False, for example A=[2 2] cannot be written as a linear combination of the two parallel vectors v1=[2;0] and v2=[1;0]; the statement however is true if you replace parallel with nonparallel vectors. #55 1.2

If a system of m linear equations in n variables is equivalent to a system of p linear equations in q variables, then m=p.

False, for example if the first system of linear equations contains twice the same row and the second system of equation only contains this equation once.

There is a unique sequence of elementary row operations that transforms a matrix into its reduced row echelon form.

False, for example multiplying a row by two is an elementary row operation, but you can obtain the same by adding the row to itself

If S_1 and S_2 are finite subsets of R^n having equal spans, then S_1 and S_2 contain the same number of vectors.

False, for example the spans S_1={[1;0] [2;0]} and S_2={[1;0]} are equal since both spans will contain the same vectors, but the first subset has two vectors, while the second subset has 1.

If S_1 and S_2 are finite subsets of R^n having equal spans, then S_1=S_2

False, for example the spans S_1={[1;0] [2;0]} and S_2={[1;0]} are equal since both spans will contain the same vectors, but the subsets are not equal.

I f A= [1 2;3 4 ] and B = [1 2 0; 1 4 0] then A = B .

False, for two matrix to be equal their size should be same here A and B are not of same size

The matrix vector product of a 2x3 matrix and a 3x I vector equals a linear combination of the rows of the matrix.

False, in the definition on page 19 we note that the can write the matrix-vector product as a linear combination of the columns and NOT the rows.

If A is an m x n matrix, then Ax = b is consistent for every b in R^m if and only if the rank of A is n.

False, in theorem 1.6 we note that this is only true

If a system of linear equations has more variables than equations, then it must have infinitely many solutions.

False, it is also possible that the system of equations has no solution.

Multiplying every entry of some row of a matrix by a scalar is an elementary row operation.

False, it is only an elementary row operation if the scalar is nonzero.

The coefficients in a linear combination can always be chosen to be positive scalars

False, look for example as exercise 43, you can only write u as a linear combination of the vectors in S using negative scalars.

If theta> 0, then A_(theta)u is the vector obtained by rotating u by a clockwise rotation of the angle theta.

False, on page 23 the rotation matrix is explained and they explicitly mention that the product is the vector obtained by a counterclockwise rotation.

The matrix vector product of an m x n matrix and a vector yields a vector in R^n

False, since A is a mxn matrix, then the vector has to be a nx1 matrix and their product is then a mx1 matrix. in other words the product is a vector in R^n.

Every matrix can be transformed into a unique matrix in row echelon form by a sequence of elementary row operations.

False, the leading entries do not have to be 1 and thus every scalar multiple of the matrix in row echelon form is also a possible row echelon form, which means that the row echelon form is not UNIQUE.

There exists a 5 x 8 matrix with rank 3 and nullity 2.

False, the nullity is the number of columns 8 decreased by the rank 3, this it is 8-3=5.

A scalar multiple of the zero matrix is the zero scalar

False, the product of a scalar and a zero matrix is again a zero matrix (NOT a zero scalar), because you multiply the scalar with each element of the matrix.

A vector with exactly one nonzero component is called a standard vector.

False, the standard vectors need to be columns of the identity matrix, and thus the vector needs to have exactly one nonzero component which has to be 1.

The sum of the rank and nullity of a matrix equals the number of rows in the matrix.

False, the sum of the rank and nullity of a matrix equals the number of columns in the matrix

The third pivot position in a matrix lies in column 3.

False, the third pivot position of a matrix always lies in the third row, but in which column it lies is not in advance determined.

If A is the coefficient matrix of a system of m linear equations in n variables, then A is an n x m matrix.

False, the variables are in columns and the linear equations in rows, thus A is an mxn matrix.

If a matrix is in row echelon form then the leading entry of each nonzero row must be 1.

False, this is not a property of matrices in row echelon form, but it is a property of matrices in reduced row echelon form.

Let S be a nonempty set of vectors in R^n, and let v be in R^n. The spans of S and S U {v} are equal if and only if v is in S.

False, v should be in the span of S instead of the S itself.

If B is a 3x4 matrix, then its rows are 4xI vectors

False. B has 3 rows and 4 columns that means that each row consists of 4 elements. Hence rows are 1x4 vectors.

The (i.j)-entry of A^T equals the (j, i)-entry of A.

In transpose column and row are interchanged so (i,j)th entry of A transpose will same as (j,i)th entry of A. True

Every vector in R^2 can be written as a linear combination of the standard vectors of R^2

Let u be any vector in R^2, then u can be written as u=[u1;u2] where u1 and u2 are scalars. we can also write u as u=[u1;;u2]=u1[1;0]+u2[0;1]. Hence every vector in R^2 is a linear combination of standard vectors in R^2.

For any m x n matrices A and B and any scalars c and d, (cA+dB)^T=cA^T+dB^T

The transposed of the sum of matrices is the sum of the transposed matrices and the transposed of the scalar multiple of a matrix is the scalar multiplied by the transposed matrix. true (cA^T)= cA^T (A+B)^T= A^T+B^T

1.1 Matrices must be the same size for their sum to be defined.

True , matrix must be of same size for there addition to be defined let A be (3x3) matrix and B be (4x4) matrix than A+B the element (3,4) will not be determined as A doesn't have (3,4)th element

If A is an m x n matrix, then a solution of the system Ax =b is a vector u in R^n, such that Au=b.

True because the solution u of the system should solve the system if you replace x with u.

The rotation matrix A_180 equals - I_2•

True rotation matrix A_180=[cos 180 -sin180; sin180 cos180]=[-1 0; 0 1]= -I_2

If A is a matrix for which the sum A +A^r is defined. then A is a square matrix.

True, is sum of A and A transpose is defined than A and A transpose are of same size And if A and A transpose are of same size than no of row =no of column therefore A is square matrix as a matrix having no of row =no of column is square matrix

If A and B are any m x n matrices. then A-B =A+(-1)B..

True, when a matrix is multiplied by a scalar each element of matrix is multiplied by that scalar So -B is same as (-1)B

A system of linear equations is called consistent if it has one or more solutions.

True, A system is consistent if there is no row in this matrix in which the only nonzero entry lies in the last column and thus if the system does not have no solutions. Since a system of linear equations has no solutions or one solution or infinitely many solutions, a system is thus consistent if it has one or more solutions.

If the only nonzero entry in some row of an augmented matrix of a system of Iinear equations lies in the last column. then the system is inconsistent.

True, A system is consistent if there is no row in this matrix in which the only nonzero entry lies in the last column; since there does lie a nonzero entry in the last column, the system thus has to be inconsistent.

In a zero matrix. every entry is 0.

True, The defination of zero matrix is that each and every entry of zero matrix is element zero. that why zero matrix is also known as null matrix

A column of a matrix A is a pivot column if the corresponding column in the reduced row echelon form of A contains the leading entry of some nonzero row.

True, a pivot column is a column that contains a leading entry in the reduced row echelon form.

If some column of matrix A is a pivot column, then the corresponding column in the reduced row echelon form of A is a standard vector.

True, because a pivot column contains a 1 on the pivot position and the rest of the column contains zeros, thus the pivot column is the column of an identity matrix. By definition of standard vectors, we then know that the pivot column has to be a standard vector.

The span of a set of two nonparallel vectors in R^2 is R^2.

True, because if you write the two nonparallel vectors in a 2x2 matrix and transform it in its reduced row echelon form, then the transformed matrix will not have any zero rows (since the two vectors were not parallel) moreover this matrix will be the identity matrix I_2 and thus the span of two nonparallel vectors is R^2.

A system of linear equations Ax = b has the same solutions as the system of linear equations Rx =c. where [R c] is the reduced row echelon form of [A b].

True, because we solve the system by finding the reduced row echelon form.

The zero vector is a linear combination of any nonempty set of vectors.

True, because you can write the zero vector as a linear combination of any (nonempty) set of vector by setting all the scalars to zero.

If a system of m linear equations in n variables is equivalent to a system of p linear equations in q variables, then p=q.

True, equivalent systems of linear equations always have the same number of variables (you can never reduce the number of variables and keep an equivalent system).

Suppose that the pivot rows of a matrix A are rows I,2..... k, and row k+I becomes zero when applying the Gaussian elimination algorithm. Then row k + I must equal some linear combination of rows I,2.....k.

True, if row k+1 becomes zero it needs to be able to become zero with only using elementary row operations and by the elementary row operations we then know that row k+1 must be linear combination of the previous rows.

If A = [u_1, u_2 . . . u_k ) and the matrix equation Ax=v is inconsistent, then v does not belong to the span of {u_1, u_2...u_k}.

True, if the matrix equation Ax=v is inconsistent, then It is not possible to write the vector v as a linear combination of the vectors in the span of A and thus v does not belong to the span of A.

Every solution of a consistent system of linear equations can be obtained by substituting appropriate values for the free variables in its general solution.

True, if you substitute the values (which are the solutions) every equation in the system should be true.

If A is a matrix with rank k, then the vectors e_1, e_2.........e_k appear as columns of the reduced row echelon form of A.

True, it immediately follows from exercise 66 (which was true).

If a matrix A can be transformed into a matrix B by an elementary row operation. then B can be transformed into A by an elementary row operation.

True, let us consider how you can return to the original matrix for every elementary row operation: If the interchange operation was used, then we can undo it by interchanging the same two rows of the matrix. If the scaling operation was used, then we can undo the operation by dividing every entry in the row by the nonzero scalar by which was multiplied. If the row addition operation was used, then you can undo the operation by subtracting the same multiple of the same row of the row that was changed.

A matrix having nonnegative entries such that the sum of the entries in each column is 1 is called a stochastic matrix.

True, look at the definition of a stochastic matrix in example 3 on page 21.

Every vector in R^2 is a linear combination of any two nonparallel vectors.

True, look at the property in the blue rectangle at the bottom of page 17.

If Ax = b is consistent, then the nullity of A equals the number of free variable; in the general solution of Ax = b.

True, on the bottom of page 49 it says that the number of free variables in a consistent system equals the nullity.

The rank of a matrix equals the number of pivot columns in the matrix.

True, see definition in blue rectangle on page 48.

If the equation Ax = b is inconsistent, then the rank of [A bJ is greater than the rank of A.

True, the augmented matrix [A b] is inconsistent if the matrix contains a row with all zeroes except for in the last column and the corresponding matrix A then has in this row all zeroes. thus we know that the rank of the augmented matrix is one greater than the rank of A

The augmented matrix of a system of linear equations contains one more column than the coefficient matrix.

True, the augmented matrix is the coefficient matrix augmented to include the vector b and thus one column is added to the coefficient matrix.

For any vector u in R^2, A_(theta)u is the vector obtained by rotating u by the angle theta.

True, the concept of the rotation matrix is explained on page 23 where they state the exact same thing.

When the forward pass of Gaussian elimination is complete. the original matrix has been transformed into one in row echelon form.

True, the forward pass transforms the matrix in the row echelon form, while the backward pass further transforms it into the reduced row echelon form.

No scaling operations arc required in the forward pass of Gaussian elimination.

True, the scaling operations are required in the backward pass of the Gaussian elimination

Every finite subset of R^n is contained in its span.

True, the span contains all vectors that are linear combinations of of the vectors in the span and thus the vectors from the span itself, always lie in the span too.

The third pivot position in a matrix lies in row 3.

True, the third pivot position of a matrix always lies in the third row, but in which column it lies is not in advance determined.

Let S = {u_1,u_2....u_k} be a subset of R^n. Then the span of S is R^n if and only if the rank of [u_1, u_2....u_k] is n.

True, this follows directly from theorem 1.6 on page 70.

Every vector v in R^n can be written as a linear combination of the standard vectors, using the components of v as the coefficients of the linear combination.

True, this is a consequence of the property I_nv=v or matrix vector products and the definition of standard vectors (which are the columns of the identity matrix I_n)

If S_1 and S_2 are finite subsets of R^n such that S_1 is contained in Span S_2, then Span S_1 is contained in Span S_2 •

True, this is a consequence of theorem 1.7 on page 72.

1.6 Let S=[u_1, u_2,......uk] be a nonempty set of vectors in R^n. A vector v belongs to the span of S if and only if v = c_1u_1+c_2u_2+···+c_ku_k for some scalars c_1, c_2,...... c_k.

True, this is a direct consequence of the definition of the span on page 66

Performing an elementary row operation on the augmented matrix of a system of Iinear equations produces the augmented matrix of an equivalent system of linear equations.

True, this is also mentioned in the second paragraph 33. The elementary row operations produce equivalent systems of equations, because the both systems of equation have the same solution set.

If a matrix is in reduced row echelon form, then the leading entry of each nonzero row is 1.

True, this is condition in the definition of the reduced row echelon form (see page 33).

If R is an n x n matrix in reduced row echelon form that has rank n then R = I_n.

True, this is mentioned in the blue rectangle on the middle of page 48.

Every matrix can be transformed into one in reduced row echelon form by a sequence of elementary row operations.

True, this is mentioned in the explanation of the reduced row echelon form on page 33.

The equation Ax = b is consistent if and only if b is a linear combination of the columns of A.

True, this is stated in theorem 1.5 on page 50.

The matrix vector product of an m x n matrix A and a vector u in R^n equals u_1a_1+u_2a_2+.....+u_na_n

True, this is the definition of the matrix-vector product on page 19.

If A is an mxn matrix, u is a vector in R^n and c is a scalar, then A(cu) = c(Au).

True, this is the second properties of matrix vector products of theorem 1.3.

Every matrix can be transformed into a unique matrix in reduced row echelon form by a sequence of elementary row operations.

True, we know that a matrix can always be transformed in reduced row echelon form (see exercise 62) and such a matrix also has to be unique since the leading entries of every row has to equal 1 (and thus the other elements in the row cannot differ).

If the reduced row echelon form of the augmented matrix of a consistent system of m linear equations in n variables contains k nonzero rows, then its general solution contains k basic variables.

True, you can determine one variable from each nonzero row, such a variable is called a basic variable and thus there has to be k basic variables in this case.

The span of {0) is {0).

True. the span of {0} contains all vectors that can be written as a linear combination of the vectors in the span. Since for every scalar k we have k0=0 and also 0+0 the only vectors that can be written as a linear combination of the vectors in the span is 0.

1.2 A linear combination of vectors is a sum of scalar multiples of the vectors.

a linear combination of vectors is the sum of the scalar multiples of these vectors (scalar is also allowed to be zero and negative) True

Every vector is a matrix

any vector can either be represented as column vector or row vector so true

The transpose of a matrix is a matrix of the same size

false , the transpose of a matrix is not of same size For ex. if A is (3x4) matrix than A transpose is (4x3) matrix. If a matrix is square matrix i.e. no of row is equal to no of column than size of matrix and its transpose is equal

An m x n matrix has m + n entries.

false A m*n matrix will have mn entries not (m+n) entries

The (3, 4)-entry of a matrix lies in column 3 and row 4.

false. the (3,4)th entry will not lie in 3rd column and fourth row ,it will lie in 3rd row and fourth column

If A is a matrix, then cA is the same size as A for every scalar c.

if multiplication by scalar does not increase no of element in a matrix

In any matrix A, the sum of the entries of 3A equals three times the sum of the entries of A.

true when a matrix is multiplied by a scalar each element of matrix is multiplied by that scalar So when sum of entry of 3A ,each element is multiplied by 3 .So 3A=3 a11+3a12+....=3(a11+a12...)=3*sum of entry of A.

Matrix addition is associative.

true, part of definition

Matrix addition is commutative.

true, part of definition

If' v and w are vectors such that v=-3w, then v and w are parallel.

when a vector is represented as some scalar time other vector than these vectors are parellel here v=-3w , -3 is scalor so v and w vector are parallel. more generally if scalar is +ve than vector are said to be parallel and when scalar is -ve than vectors are said to be anti-parallel. true

A submatrix of a matrix may be a vector.

yes, the submatrix of a matrix may be vector as if we choose any column of matrix it will be column vector and if we choose any row of a matrix it will be row vector. true