ULTIMATE Linear Algebra 33A Studyset
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
