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MATRICES TRUE/FALSE QUESTIONS AND ANSWERS
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MATRICES TRUE/FALSE QUESTIONS AND ANSWERS
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MATRICES TRUE/FALSE QUESTIONS AND ANSWERSAny linear combination of vectors can always
be written in the form Ax for a suitable
matrix A and vector x. (HW 01) - Answers -True
The determinant of a triangular matrix is
always the sum of the entries on the main
diagonal. (HW 01) - Answers -False
If v is a vector in a vector space V , then
the scalar multiple (−1)v is the same as the
negative of v. (HW 03) - Answers -True
A vector space is infinite-dimensional if it
is spanned by an infinite set. (HW 04) - Answers -False
If B = {v1, v2, . . . , vp} is a basis for a subspace H of Rn and if x = c1v1 + c2v2 + · · · +
cpvp for a vector x in H, then c1, c2, . . . , cp are the coordinates of x relative to the
basis B. (HW 04) - Answers -True
If B = {v1, . . . , vp} is a basis for a subspace H of Rn, then the correspondence x 7→
[x]B makes H look and act the same as Rp. (HW 04) - Answers -True
The columns of an invertible n × n matrix
form a basis for Rn. (HW 04) - Answers -True
If A is a 4 × 5 matrix, then
dim(Col(A)) + dim(Nul(A)) = 4. (HW 04) - Answers -False
The dimension of Nul A is the number of
variables in the equation Ax = 0. (Exam 01) - Answers -False
The set H of all polynomials
p(x) = 3 + bx^5, b in R, is a subspace of the vector space P6 of all polynomials of
degree at most 6. (Exam 01) - Answers -False
f A is an m × n matrix and if the equation
Ax = b is inconsistent for some b in Rm, then
A cannot have a pivot position in every row. (Exam 01) - Answers -True
When A is a 3 × 7 matrix, the smallest
possible dimension of the null space of A is 4. (Exam 01) - Answers -True
A consistent system of linear equations has
, one or more solutions. (HW 02) - Answers -True
Elementary row operations on an augmented
matrix never change the solution set
of the associated linear system. (HW 02) - Answers -True
If the equation Ax = b has at least two
different solutions, then it has infinitely many
solutions. (HW 02) - Answers -True
Given an invertible 5 × 5 matrix A and
a vector b in R5, then the matrix equation
Ax = b has a unique solution. (HW 02) - Answers -True
If matrices A and B are row equivalent,
they have the same reduced row echelon form. (HW 02) - Answers -True
If an augmented matrix [A b] is transformed
into [C d] by elementary row operations,
then the linear systems they represent
have exactly the same solution set. (HW 02) - Answers -True
A set H is a subspace of Rn when
1. the zero vector is in H,
2. u, v are in H when u + v is in H,
3. cv is in H for each v in H and scalar c. (HW 03) - Answers -False
For each fixed 3 × 2 matrix B, the corresponding
set H of all 2 × 4 matrices A such
that BA = 0 is a subspace of R2×4. (HW 03) - Answers -True
The set H of all polynomials
p(x) = a + bx^2, a, b in R, is a subspace of the vector space P6 of all polynomials of
degree at most 6. (HW 03) - Answers -True
Given vectors v1, v2, . . . , vp in Rn, the set of all linear combinations x1v1 + x2v2 + . . .
+ xpvp of these vectors is a subspace of Rn. (HW 03) - Answers -True
If f is a function in the vector space V of all
real-valued functions on R and if f(t) = 0 for
some t, then f is the zero vector in V. (HW 03) - Answers -False
A subspace H of a vector space V is a vector
space by itself. (HW 03) - Answers -True
If a set {v1, . . . , vp} spans a finitedimensional
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