Winter 2020 Math 21 Practice Midterm
1) – 3) Determine if the linear system with the given augmented matrix has no solutions (inconsistent), a unique solution, innitely many solutions.
Question 1.The system
5x − y + z = 0 4x−3y+7z =0.
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Question 2. The augmented matrix of a linear system L is row equivalent to
the matrix
1 2 −2 5 |0 01 4 2|2. 0 0 1 3 |3
0 0 0 0 |1
Question 3. The augmented matrix of a linear system L is row equivalent to
the matrix
13 5 7|1 0 1 −4 8 |3. 0 0 1 9 |8
0 0 0 1 |4
Question 4 Find a general solution to the matrix equation:
1 −2 1 x1 −4 0 1 2x2=−1 −131×3 3
Question 5. Let A = 1 1 1. For what values of h the columns of A
2 −1 0 are linearly independent?
Question 6. Consider the following basis for R4:
1 0 0 1
1 1 0 −1 v1 =0,v2 =1,v3 =1,v4 = 1 .
Find the coordinate vector 2
4 Question 7. Let v = 0
of v = 3 with respect to basis {v1,v2,v3,v4}. 5
and w = 2. Find Proj ⊥ (w). 2 v
Winter 2020 Math 21 Practice Midterm
Question 8. Let T : R3 → R3 be a linear transformation such that 1 2 0 1 0 0 2
T 0 = 4 , T 1 = 3 , T 0 = −2. Find T 3.
0 −1 0 −2 1 2 1 3 4 −6
002 and let W = Span (v1, v2, v3). Determine W ⊥.
1 1 1 0 Inquestions13and14,letA=0 3 1 2.
andB=4 −2 6. 2451 572
Inquestions9. and10. letA= 5 −1 4 2
811
Question 9. Determine the size of the matrix BA. Question 10. Determine the (2, 3)-entry of AB.
1 −2 1 −1 Question 11. Find the null space of the matrix A = 1 1 1 −1.
Question 12. Let
1 1 1 0 2 0
v1 = 2,v2 = 0,v3 = 0
2 3 1 0 1021
Question 13. Find A−1.
Question 14. Find the sum of the main diagonal entries of A−1.
1 2 3 0 1 0 1 0 −1 2 3 Question15.LetA=4 5 6 −1,B= 2 −1 0 ,C= 1 1 6 4 .
Calculate A+2BTrC.
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