IIITDM KANCHEEPURAM
MAT204T: Linear Algebra
Supplementary Examination
July 20, 2021
9:30am – 12:00pm
Answer All Questions
Marks: 50
1. Solve the following system by Gaussian elimination method.
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x1 − 2x2 + x3 − 4x4 = 1
x1 + 3x2 + 7x3 + 2x4 = 2
x1 − 12x2 − 11x3 − 16x4 = 5
2. Prove that any set of m vectors in Rn , where m > n, is linearly dependent.
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3. Let C = AB, where A, B are n × n square matrices. Prove that if C is invertible, then
A and B are both invertible.
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4. Determine whether the subspaces of R4 spanned by A = {(1, 0, 1, 0), (0, 1, 2, 0), (0, 0, 0, 1)}
and B = {(1, 0, 2, 0), (0, 1, 1, 0), (0, 0, 0, 1)} are equal.
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5. Suppose you are given m linearly independent vectors in Rn , where m < n. Provide a
simple method for finding a basis for Rn that contains the given vectors.
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6. Let T : R2 → R2 be the reflection of the plane about the line through the origin and
the point (1, 2). Find a formula for T (x, y). Show that it is a linear transformation. (4)
7. State and prove the dimension theorem (rank-nullity theorem).
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8. Find a linear transformation T : R3 → R3 such that its range RT = {(x, y, z) |
x + y + z = 0} and its null space NT = {(t, 2t, 3t) | t ∈ R}.
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9. Let T : R2 → R2 be defined by T (x, y) = (3x, 5y). Find its matrix representation in
the ordered bases (i) β = ((1, 0), (0, 1)) and (ii) β 0 = ((1, 1), (1, −1)).
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10. Diagonalize the following matrix:
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

5 −6 −6
2
A = −1 4
3 −6 −4
11. Prove that if v1 , v2 , . . . , vn are nonzero orthogonal vectors in an inner product space,
then they are linearly independent.
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12. Consider the vector space R3 equipped with the standard inner product. Apply the
Gram-Schmidt process to w1 = (3, 0, 4), w2 = (−1, 0, 7), w3 = (2, 9, 11).
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