Practice Problem for Midterm #2, Math 1502
1. Determine whether the given vectors are linearly independent or dependent. If the vectors are
linearly dependent, express 0 as a nontrivial linear combination of the vectors:
0 1 0 1 0 1
1 2 -3
a)@ -2A ;@ -5A ; @ 2 A
3 8 -1
0 1 0 1 0 1
1 1 0
b)@ 1 A; @ 0 A ;@ 1 A
1 1 1
▯ ▯ ▯ ▯ ▯ ▯
c) 1 ; 1 ; -3
-2 -5 2
0 1 0 1
2 -2
B C B C
d)B -2C ;B 2 C
@ 3 A @ -3A
4 4
2. For every system of equations given below perform the following steps:
(a) Write the augmented matrix of the system.
(b) Reduce the system to reduced echelon form.
(c) Find the solutions of the system.
(d) Express the solution in vector form.
a)
▯2x2+ x3▯ x4= 3
x + x ▯ 2x + 3x = 2
1 2 3 4
▯x1+ 3x3= ▯1
b)
x1+ x2+ 4x3+ 3x4= 0
2x + 4x + 9x ▯ x = 0
1 2 3 4
c)
x1+ x2+ x 3 5
▯x 1 2x 2 2x 3 1
3x1▯ x 2 5x 3 3
c)
x1+ x2= 0
▯4x + 3x = 1
1 2
▯3x 1 4x 2 ▯1
1 3. Find the matrix of a linear transformation T given the information below:
▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯
(a) T 1 = 3 ; T 4 = 1
0 4 4 -3
0 1 0 1 0 1 0 1 0 1 0 1
0 2 0 1 1 2
@ 0 A @ 3A @ 1A @ -3A @ 1A @ 2 A
(b) T = ; T = ; T =
1 -2 1 1 1 1
4. ▯ ▯ ▯ ▯ ▯ ▯
1 1 2
(a) Is-3 in the span -1 and -3 ? If yes, ▯nd a speci▯c linear combination.
0 1 0 1 0 1 0 1
1 1 0 2
(b) Is -3A in the span of1A , -2A

[email protected] 0 A? If yes, ▯nd a speci▯c linear
-2 2 1 1
combination.
0 1 0 1 0 1
1 2 3
(c) Is 1A in the span of2A

[email protected] -1A ? If yes, ▯nd a speci▯c linear combination.
-2 3 1
5. For a linear transformations T ▯nd the following:
0 1 0 1 0 1 0 1 0 1
1 2 0 0 1
@ A @ A @ A @ A @ A
(a) Given that T = 1 and T 1 = 0 ▯nd T -1 .
1 -1 1 2 0
0 2 1 0 1 1 0 6 1
(b) Given that T3A = @ -5A ▯nd

[email protected] -9A.
1 6 3
0 1
0
(c) With no information, ▯nd.T
0
6. At a store a notepad costs $2 and a notebook costs $3. In a day 110 items were sold and $264
were spent. Find how many notepads were sold.
7. True or False. No partial credit.
(a) The span of the columns of a matrix A is equal to the image of the linear transformation
T given by T(x) = Ax.
(b) Any 3 v