Linear Algebra Exam 1 UH

a mapping T : R^n -> R^m is onto R^m if every vector x in R^n maps onto some vector in R^m (t/f)

false

a mapping T: R^n --> R^m is one to one if each vector in R^n maps onto a unique vector in R^m (t/f)

false

every linear transformation is a matrix transformation (t/f)

false

if A is a 3x2 matrix , then the transformation x -> Ax cannot be one to one (t/f)

false

if A is a 4 x 3 matrix , then the transformation x -> Ax maps r^3 onto R^4 (t/f)

false

if a set in R^n is linearly dependent , then the set contain more than n vectors. (t/f)

false

the standard matrix of a horizontal shear transformation from R^2 to R^2 has the form a 0 under 0 d , where a and d are plus/minus 1.

false

two equivalent linear systems can have a different solution sets. (t/f)

false

when two linear transformation are performed one after another , the combined effect may not always be a linear transformation (t/f)

false

when u and v are non zero vectors , span (u,v) contains only the line through u and the origin , and the line through v and the origin(t/f)

false , Span{u, v} can be a plane.

the set span (u , v) is always visualized as a plane through the origin. (t/f)

false , The statement is often true, but Span{u, v} is not a plane when v is a multiple of u, or when u is the zero vector.

a linear transformation T: R^n -> R^m is completely determined by its effects on the columns of the n x n identity matrix (t/f)

true

a linear transformation is a special type of function ? (t/f)

true

if Ax=b is consistent , then the solution set of Ax=b is obtained by translating the solution set of Ax=0

true

if T : R^2 -> R^2 rotates vectors about the origin through an angle theta , then T is a linear transformation (t/f)

true

if three vectors in R^3 lie in the same plane in R^3, then they are linearly dependent (t/f)

true

if u and v are linearly independent, and if w is in span {u,v} then {u,v,w} is linearly dependent. (t/f)

true

the columns of the standard matrix for a linear transformation from R^n to R^m are the images of the columns of the n x n identity matrix under T (t/f)

true

the effect of adding p to a vector is to move the vector in the direction parallel to p

true

the first entry in the product Ax is a sum of products (t/f)

true

the range of the transformation x -> Ax is the set of all linear combinations of the columns of A (t/f)

true

the reduced echelon form of a matrix is unique (t/f)

true

the solution set of a linear system whose augmented matrix is [ a1 a2 a3 b] is the same as the solution set of the equation x1a1+x2a2+x3a3=b (t/f)

true

a basic variable of a linear system is a variable that corresponds to a pivot column in the coefficient matrix(t/f)

true , basic variable are defined after equation (4)

a linear transformation T : R^n -> R^m always maps the origin of R^n to the origin of R^m (t/f)

true T(0)=0

the equation ax=b is homogenous if the zero vector is a solution

true if the zero vector is a solution then b = Ax=A0=0

the columns of any 4x 5 matrix are linearly dependent (t/f)

true see fig 3 after theorem 8

the solution set of a linear system whose augmented matrix is [ a1 a2 a3 b} is the same solution set of Ax=b , if A [ a1 a2 a3 ]

true theorem 3

a consistent system of linear equations has one or more solutions (t/f)

true, consistent system has at least one solution.

theorem 8

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1,...,vp} in Rn is linearly dependent if p>n.

chapter 2 theorem 3

Let A and B denote matrices whose sizes are appropriate for the following sums and products. a) (AT)T = A. b) (A+B)T = AT + BT. c) For any scalar r, (rA)T = rAT. d) (AB)T = BTAT.

theorem 1

each matrix is row equivalent to one and only one reduced echelon form matrix.

the equation Ax=b is referred to as a vector equation (t/f)

false , it is a matrix equation.

Another notation for the vector [-4 3] is [-4 3] t/f

false , the alternative notation for a column vector is (-4,3) using parentheses and commas.

if S is a linearly dependent set, then each vector is a linear combination of the other vector in S (t/f)

false . see warning after theorem 7

a 5x6 matrix has 6 rows (t/f)

false, a 5x6 matrix has 5 rows and 6 columns

the row reduction algorithm applies only to augmented matrices for a linear system.(t/f)

false.

if x is a non trivial solution Ax=0 , then every entry in x is non zero. (t/f)

false. A non trivial solution of Ax=0 is any non zero x that satisfies the equation.

if A is an m x n matrix whose columns do not span R^m , then the equation Ax=b s consistent for every b in R^m

false. in theorem 4, state (c) is false if and only if statement (a) is also false.

the equation ax=b is consistent if the augmented matrix [A b] has a pivot position in every row (t/f)

false. see warning following theorem 4.

the homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable (t/f)

false. the equation Ax=0 always has the trivial solution

linear dependent

if 1. there exist weights 2. there is a whole zero vector/ matrix. 3. contains more vectors than there are entries in a single vector.

theorem 3

if A is an m x n matrix , with columns a1... an and if b is in r^m , the matrix equation Ax=b has the same solution set as the vector equation x1a1 +x2a2+...xnan=b which in turn, has the same solution set as the linear system whose augmented matrix is [ a1 a2 ... an b]

homogenous

if it can be written in the form Ax=0

inconsistent

if it has no solution

interchange (elementary row operations)

interchange two rows

Linear System

is a collection of one or more linear equation involving the same variables

singular matrix

is an not an invertible matrix

scaling (elementary row operations)

multiply all entries in a row by a non zero constant.

non singular matrix

one that is invertible

leading entry

refers to the leftmost entry in a non zero row.

a linear transformation preserves the operations of vector addition and scalar multiplication (t/f)

true

any list of five real numbers is a vector in r ^5 (t/f)

true

asking whether the linear system corresponding to an augmented matrix [ a1 a2 a3 b] has a solution amounts to asking whether b is in span { a1, a2 , a3} (t/f)

true

elementary row operations on an augmented matrix never change the solution set of the associated linear system. (t/f)

true

every linear transformation from R^n to R^m is a matrix transformation (t/f)

true

every matrix transformation is a linear transformation (t/f)

true

an example of a linear combination of vectors v1 and v2 is the vector 1/2 v1. (t/f)

true.

every elementary row operation is reversible (t/f)

true.

two fundamental questions about a linear system involve existence and uniqueness. (t/f)

true.

if the columns of an m x n matrix A span R^m , then the equation Ax=b is consistent for each b in R^m (t/f)

true. see part c and part a in theorem 4.

if x and y are linearly independent and if {x,y,z} is linearly dependent , then z is in span {x,y} (t/f)

true. see remark following example 4.

A system of linear equations has

- no solution , or -exactly one solution , or - infinitely many solutions.

theorem 9

If a set S = {v1,...,vp} in Rn contains the zero vector, then the set is linearly dependent.

finding a parametric description of the solution set of a linear system is the same as solving the system (t/f)

True.

row echelon form

1. all non zero rows are above any row of all zeros 2. each leading entry of a row is in a column to the right of the leading entry of the row above it 3. all entries in a column below a leading entry are zeros.

writing a solution set (of a consistent system) in parametric vector form

1. row reduce the augmented matrix to reduced echelon form. 2. express each basic variable in terms of any free variables appearing in an equation. 3. write a typical solution x as a vector whose entries depend on the free variables, if any 4. decompose x into a linear combination of vectors (with numeric entries) using the free variable as parameters.

reduced row echelon form

4. the leading entry in each nonzero row is 1 5. each leading 1 is the only non zero entry in its column.

theorem 7

An indexed set S = {v1,...,vp} of two or more vectors is linearly dependent IFF at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v1 ≠ 0, then some vj (j>1) is a linear combination of the preceding vectors v1,...,v(j-1)

if a set contains a few vectors than there are entries in the vectors, then the set is linearly independent. (t/f)

False.

the vector v results when a vector u-v is added to the vector v

False. (u - v) + v = u - v + v = u.

the pivot position in a matrix depend on whether row interchanges are used in the row reduction process. (t/f)

False. See the beginning of the subsection Pivot Positions. The pivot positions in a matrix are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix.

the weights c1....cp in a linear combination c1v1 +... cpvp cannot all be zero

False. Setting all the weights equal to zero results in a legitimate linear combination of a set of vectors.

the solution set of a linear system involving variables x1...xn is a list of numbers (s1... sn) that makes each equation in the system a true statement when the values s1...sn are substituted for x1....xn (t/f)

False. The description applies to a single solution. The solution set consists of all possible solutions. Only in special cases does the solution set consist of exactly one solution. Mark a statement True only if the statement is always true.

whenever a system has a free variables, the solution set contains many solutions

False. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty. See the solution of Practice Problem 2.

is A in an m x n matrix, then the range of the transformation x to Ax is R^m (t/f)

False. The range is the set of all linear combinations of the columns of A. See the paragraph before Example 1.

if one row in an echelon form of an augmented matrix is [0 0 0 5 0 ] , then the associated linear system is inconsistent. (t/f)

False. The row shown corresponds to the equation 5x4 = 0, which does not by itself lead to a contradiction. So the system might be consistent or it might be inconsistent.

the solution set of Ax=b is the set of all vectors of the form w = p + Vh where Vh is any solution of the equation Ax=0 (t/f)

False. The solution set could be empty! The statement (from Theorem 6) is true only when there exists a vector p such that Ap = b.

a homogenous system of equations can be inconsistent. (t/f)

False. The trivial solution is always a solution to a homogeneous system of linear equations.

theorem 5

If A is an invertible square matrix, then for each b in Rn, the equation Ax = b has the unique solution x = A^(-1)b.

theorem 5

If A is an m x n matrix, u and v are vectors in Rn, and c is a scalar, then a) A(u+v) = Au + Av, b) A(cu) = c(Au)

theorem 4

Let A = [ab<br />cd]. If ad-bc ≠ 0 then A is invertible and A^(-1) = 1/(ad-bc)[d -b <br /> -c a]. If ad-bc = 0, A is not invertible.

theorem 4

Let A be an m x n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. a) For each b in R^m, the equation Ax=b has a solution. b) Each b in R^m is a linear combination of the columns of A. c) The columns of A span R^m d) A has a pivot position in every row *only about coefficient matricies

chapter theorem 2

Let A be an mxn matrix, and let B and C have sizes for which the indicated sums and products are defined. a) Associative law of multiplication: A(BC)=(AB)C. b&c) Distributive laws. d) r(AB) = (rA)B=A(rB) for any scalar r. e) Identity for matrix multiplication: ImA = A = AIn.

chapter 2 theorem 1

Let A,B,C be matrices of same size, let r and s be scalars. a) A+B=B+A. b) (A+B)+C =A+(B+C). c) A+0=A. d) r(A+B)=rA+rB. e) (r+s)A = rA+sA. f) r(sA) = (rs)A

theorem 12

Let T : Rn -> Rm be a linear transformation and let A be the standard matrix for T. Then: a) T maps Rn onto Rm IFF the columns of A span Rm; b) T is 1-to-1 IFF the columns of A are linearly independent.

theorem 11

Let T : Rn -> Rm be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution.

theorem 10

Let T: Rn -> Rm be a linear transformation. Then there exists a unique matrix A s.t. T(x) = Ax for all x in Rn. In fact, A is the mxn matrix whose jth column is the vector T(ej), where ej is the jth column of the identity matrix in Rn. A= [T(e1) ... T(en)]

theorem 6

Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p + vh, where vh is any solution of the homogenous equation Ax = 0.

any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x. (t/f)

True. See Example 2.

if the equation Ax=b is consistent, then b is in the spanned by the columns of A. (t/f)

True. See the box before Example 3.

a general solution of a system is an explicit description of all solutions of the system.

True. See the paragraph just before Example 4.

every matrix equation Ax=b corresponds to a vector equation with the same solution set. (t/f)

True. This statement is in Theorem 3. However, the statement is true without any "proof" because, by definition, Ax is simply a notation for x1a1 + ⋅ ⋅ ⋅ + xnan, where a1, ..., an are the columns of A.

theorem 2

a linear system is consistent if an only if the rightmost column of the augmented matrix is not a pivot column - that is , if an only if an echelon form of the augmented matrix has no row of the form [0....0 b] with b non-zero. if a linear system is consistent , then the solution set contains either a unique solution , when there are no free variables, or (ii) infinitely many solutions , when there is at least on free variables.

theorem 6

a. If A is an invertible matrix, then A^(-1) is invertible and (A^(-1))^(-1) = A. b. If A and B are square matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is, (AB)^(-1) = B^(-1) A^(-1). c. If A is an invertible matrix, then so is AT and the inverse of AT is the transpose of A^(-1). That is, (AT)^(-1) = (A^(-1))T

the points on the plane corresponding ( -2, 5) and (-5, 2) lie on a line through the origin (t/f)

false . Plot the points to verify this. Or, see the statement preceding Example 3. If −5 were on the line through −2 and the origin, then −5 would have to be a multiple of −2 , which is 5 2 5 not the case.

the columns if a matrix A are linearly independent if the equation Ax=0 has the trivial solution. (t/f)

false a homogenous equation always has a trivial solution.

if A is a 3x5 matrix and T is a transformation defined by T(x)= Ax, the the domain of T is R^3 (t/f)

false the domain is R^5

if T : R^n -> R^m is a linear transformation and if c is in R^m then a uniqueness question is " is c in the range of T?" (t/f)

false the question is an existence question

if every column of an augmented matrix contains a pivot, the the corresponding system is consistent. (t/f)

false theorem 2

if the coefficient matrix A has a pivot position in every row, then the equation Ax=b is inconsistent (t/f)

false theorem 4 statement d is true if and only if a is true

two matrices are row equivalent if they have the same number of rows. (t/f)

false, the definition of row equivalent requires that there exist a number of row operations that transforms one matrix into the other.

the equation Ax=0 gives an explicit description of its solution set. (t/f)

false. the equation Ax=0 gives an implicit description of its solution set.

the equation x=p+tv describes a line through v parallel to p (t/f)

false. the line goes through p parallel to v

in some cases , a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. (t/f)

false. theorem 1

Consistent

if it has one or infinitely many solutions

linear independent

if there are no weights

replacement (elementary row operations)

replace one row by the sum of its self and a multiple of another row.

if A is an m x n matrix and if the equation Ax=b is inconsistent for some b in R^m , then A cannot have a pivot position in every row. (t/f)

true, theorem 4 statement (a) if false if and only if statement (d) is also false

a homogenous equation is always consistent. (t/f)

true.

a vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution. (t/f)

true.