ULTIMATE Linear Algebra 33A Studyset

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A function T from R^m to Rⁿ is called a ________

linear transformation

The number of solutions of a matrix is called the _____.

rank

If two matrices A and B have the same reduced row echelon form, then the equations Ax = 0 and Bx = 0 must have the same solutions.

T

If v and w are vectors in R⁴, then the zero vector R⁴ must be a linear combo of v and w.

T

If v and w are vectors in R⁴, then v must be a linear combo of v and w.

T

If vector u is a linear combo of vectors v and w, and v is a linear combo of vectors p, q, and r, then u must be a linear combo of p, q, r, and w.

T

If vector u is a linear combo of vectors v and w, the we can write u = av + bw for some scalars a and b

T

In Ax = b, b is a linear combination of x.

T

Matrix -.6 .8 -.8 -.6 represents a rotation

T

Matrix 1 1 0 1 0 1 1 1 0 is invertible

T

Matrix 120 001 000 is in rref

T

The equation A²=A holds for all 2 x 2 matrices A representing a projection.

T

The equation A⁻¹ = A holds for all 2 x 2 matrices A representing a reflection

T

The formula (A²)⁻¹ = (A⁻¹)² holds for all the invertible matrices A.

T

The linear system Ax=b is consistent if (and only if) rank(A) = Rank[A|b]

T

The matrix 5 6 ⁻6 5

T

The matrix product a b c d * d -b -c a is always Iⁿ

T

There exists a 2 x 2 matrix A such that A[¹₂] = [³₄]

T

There exists a matrix A such that 1 2 1 2 *A = 1 1 1 1

T

There exists a positive integer n such that 0 -1 1 0 =I²

T

There exists a real number k sucht that the matrix (k-1) -2 -4 ((k-3) fails to be invertible

T

There exits an upper triangular 2 x 2 matrix A such that A² = 1 1 0 1

T

matrix (k) -2 5 (k-6) is invertible for all real numbers k.

T

rank 111 123 136 is 3

T

there exists a matrix A such that A[⁻¹₂] = 3 5 7

T

There exists a nonzero upper triangular 2 x 2 matrix A such that A² = 0 0 0 0

T A = 0 b 0 0

The rank of any upper triangular matrix is the number of non zero entries on its diagonal.

F

The system Ax = 0 0 0 1 is inconsistent for all 4x3 matrices A

F

The system Ax = b is inconsistent if (and only if) rref(A) contains a row of zeros.

F

There exist scalars a and b such that matrix 0 1 a -1 0 b -a -b 0 has rank 3.

F

There exists a 3 x 4 matrix with rank 4.

F

There exists a 4 x 3 matrix A of rank 3 such that A* 1 2 3 equals 0

F

There exists a 5 x 5 matrix A of rank 4 such that the system Ax=0 has only the solution x = 0

F

There exists a matrix A such that A* 1 1 1 1 = 1 2 1 2

F

There exists a system of three linear equations with three unknowns that has exactly three solutions.

F

There exists an invertible 2 x 2 matix A such that A ⁻¹ = 1 1 1 1

F

There exists an invertible n x n matrix with two identical rows.

F

if matrix a b c d e f g h i is invertible then so is a b d e

F

rank 2 2 2 2 2 2 2 2 2 is 2

F

there exists a 2 x 2 matrix A such that A*[¹₁] = [¹₂] and A[²₂]= [²₁]

F

the function T[x;y] = [y;1] is a linear transformation

F if you plug in 0,0, then it would equal 0;1

There exists a real number k such that the matrix (k-2) 3 -3 (k-2) fails to be invertible

F, Note that det(A)=( k − 2)2+ 9 is always positive, so that A is invertible for all values of k

matrix (1/2) (-1/2) (1/2) (1/2)

F, The columns of a rotation matrix are unit vectors

The formula det(2A) = 2det(A) holds for all 2 x 2 matrices A.

F

The matrix 1 1 1 ⁻1

F

If A is a 3 x 4 matrix and B is a 4 x 5 matrix, then AB will be a 5 x 3 matrix.

F, Matrix AB will be 3 × 5

The number of free variables is equal to ______

m - rank(A)

n x m matrix

rows by columns

If vector u is a linear combo of vectors v and w, the w must be a linear combo of u and v.

F

Matrix 1 2 3 6 is invertible

F

The formula AB = BA holds for all n x n matrcies A and B.

F

A linear system with fewer unknowns than equations must have infinitely many solutions or none.

F

A system of four linear equations in three unknowns is always inconsistent

F

If A and B are any two 3 x 3 matrices of rank 2, then A can be transformed into B by means of rref.

F

If A and B are matrices of the same size, then the formula rank(A+B) = rank(A) + rank(B) must hold.

F

If A²=I₂, then matrix A must be either I₂ or -I₂

F

If A¹⁷ = I₂, the matrix A must be I₂

F

If matrices A and B are both invertible, then matrix A + B must be invertible as well.

F

If matrix A is in rref, the at least one of the entreis in each column must be 1

F

If matrix E is in rref, and if we omit a column of E, the the remaining matrix must be in rref

F

If the system Ax=b has a unique solution, then A must be a square matrix

F

If u, v, and w are nonzero in R², then w must be a linear combo of u and v.

F

Rank of a matrix is __________.

Number of leading 1's

R² to R

R² to R Two inputs (x) and one output (y) in a linear transformation Rⁿ to Rⁿ refers to number of input to number of ouputs

A= 1 k 0 1 A³ = 1 3k 0 1 for all real numbers k

T

If A and B are any two n x n matrices of rank n, then A can be transformed into B by means of rref ops.

T

If A and B are two 2 x 2 matrices such that the equations Ax = 0 and Bx = 0

T

If A and B are two 4 x 3 matrices such that Av=Bv for all vectors v in R³, then matrices A and B must be equal.

T

If A is a 3 x 4 matrix and vector v is in R⁴, then vector A*v is in R3.

T

If A is a 4 x 3 matrix of rank 3 and Av = Aw for two vectors v and w in R³, then vectors v and w must be equal.

T

If A is a 4 x 4 matrix and the system Ax = 2 3 4 5 has a unique solution, the the system Ax = 0 has only the solution x = 0.

T

If A is a nonzero matrix of the form a -b b a then the rank of A must be 2

T

If A is a x 4 matrix of rank 3, the the system Ax = 1 2 3 must have infinitely many solutions

T

If A is an n x n matrix and x is a vector in Rⁿ, then the product Ax is a linear combo of the columns of the matrix A

T

If A is any 4 x 3 matrix, then there exists a vector b in R⁴ such that the system Ax=b is inconsistent.

T

If A is any invertible n x n matrix, then A commutes with A⁻¹.

T

If A is any invertible n x n matrix, then rref(A) = Iⁿ.

T

If AB = Iⁿ for two n x n matrices A and B, the A must be the inverse of B.

T

If A² = A for an invertible n x n matrix A, then A must be Iⁿ.

T

If A² = Iⁿ, then matrix A must be invertible.

T

If A² is invertible, the matrix A itself must be invertible.

T

If Matrix E is in rref, and if we omit a row of E, then the remaining matrix must be rref form as well.

T

If a vector v in R⁴ is a linear combo of u and w, and if A is a 5 x 4 matrix, then Av must be linear combo of Au and Aw.

T

If matrices A and B commute, then the formula A²B = BA² must hold.

T

If matrix A is invertible, then matrix 5A must be invertible as well.

T

If the 4 x 4 matrix A has rank 4, then any linear system with coefficient matrix A will have a unique solution.

T

The column of a matrix is called a __________.

Vector

If A is an n x n matrix, x and y are vectors in R^m; and k is a scalar, then a. A(x+y) = ______ b. A(kx) = _______

a. Ax + Ay b. k(Ax)

Reduced row-echelon form

a. First non-zero entry must be a 1 b. Leading 1 has zeros in the rest of a column c. Leading 1's above must be further to the left d. Rows with all 0's must go on the bottom

Vector refers to the rows or columns of a matrix?

columns

Which is more important, rows or columns of a matrix?

columns

System has at most one solution if rank = ______

m, the # of columns

System has infinite or no solutions if rank < _____

m, the # of columns

The number of variables is equal to _______

m, the # of columns

System is consistent if rank = ______

n, the # of rows


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