linear

Question 1

Let A be an n x n matrix. The system Ax = b is consistent if every column of A has a pivot.

Answer

true

Question 2

If u, v, and w are vectors in R^3 and are linearly independent, then they span R^3.

Answer

true

Question 3

Let A be an n x n matrix. Then the system Ax = 0 has infinitely many solutions if A is invertible.

Answer

true

Question 4

If the system Ax = 0 has a non-trivial solution, then it has infinitely many solutions.

Answer

true

Question 5

If A is an n x n matrix and columns of A are linearly independent, then A is an invertible matrix.

Answer

true

Question 6

If the second row of the matrix A is a zero row, then the second row of AB is also zero.

Answer

true

Question 7

If A and B are invertible matrices, then (AB)^-1 = A^-1B^-1.

Answer

true

Question 8

Let A be an invertible matrix. Then the system Ax = b has a solution x = b/A.

Answer

true

Question 9

Let A be an invertible matrix. Then detA^-1 = 1/detA.

Answer

true

Question 10

If all the diagonal elements of an n x n matrix are zero, then det(A) = 0

Answer

true

Question 11

If A is an n x n matrix and Ax = 0 has a nontrivial solution, then the echelon form of A has less than n pivots.

Answer

true

Question 12

If {v1, v2, ..., vn} is a linearly independent set in a vector space V, the {v1, v2, ..., vn} is a basis for V.

Answer

true

Question 13

If A is a non-invertible n x n matrix, then the columns of A form a basis of R^n.

Answer

true

Question 14

The column space of an m x n matrix is R^n

Answer

true

Question 15

The number of pivot columns of a matrix equals the dimension of its column space.

Answer

true

Question 16

If v1 and v2 are eigenvectors of A, then v1 + v2 is also an eigenvector of A.

Answer

true

Question 17

The linear transformation T(x) = [-1 0 ; 0 1]x reflects a figure about the y-axis.

Answer

true

Question 18

If the n x n matrix A has n eigenvalues, then A is diagonalizable.

Answer

true

Question 19

If T is a linear transformation, the T(au + bv) = aT(u) + bT(v).

Answer

true

Question 20

Let T: R^n -> R^m be a linear transformation and {v1, v2, ..., vk} be a linearly independent set in R^n. Then {Tv1, Tv2, ..., Tvk} is a linearly independent set.

Answer

true

Question 21

If A and B are similar, then A and B have the same eigenvalues.

Answer

true

Question 22

A diagonalizable matrix is always invertible.

Answer

true

Question 23

The eigenspace of A corresponding to the eigenvalue lambda is Null(A - lambda * I)

Answer

true

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