Scribd
Upload
35 views1 page

Assignment 1

The document contains 14 multi-part math and linear algebra problems involving finding bases, spans, linear combinations, and subspaces for vector spaces and matrices. The problems cover topics like finding bases for column spaces, null spaces, and row spaces of matrices, determining if sets of vectors span vector spaces, expressing vectors as linear combinations, finding parametric equations of lines, and determining linear dependence and independence of sets of vectors.

Uploaded by

Ad K
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
35 views1 page

Assignment 1

The document contains 14 multi-part math and linear algebra problems involving finding bases, spans, linear combinations, and subspaces for vector spaces and matrices. The problems cover topics like finding bases for column spaces, null spaces, and row spaces of matrices, determining if sets of vectors span vector spaces, expressing vectors as linear combinations, finding parametric equations of lines, and determining linear dependence and independence of sets of vectors.

Uploaded by

Ad K
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 1

Assignment 1

 
1 −3 2 2 1
1. Find a basis for the column space and null space of A =  0 3 6 0 −3
−2 9 2 −4 −5

2. Let B = {1 + x, a0 + a1 x + x2 , a2 x2 } be an ordered basis for P2 . If the coordinate vector


of 2 − 3x + x2 with respect to B is (2, −5, 6), find the values of a0 , a1 , a2 .
3. Let {v1 , v2 , v3 } be a basis for a vector space V . Show that {u1 , u2 , u3 } is also a basis
for V where u1 = v1 , u2 = v1 − v2 , u3 = v1 + v2 − v3 .
4. Find a basis and dimension of the set of all 3×3 (i) upper triangular matrices, (ii)symmetric
matrices and (iii) skew symmetric matrices.
5. Which among the following are subspaces of R3 ,explain.
a. W = {X : AX = 0 where A is a 3 × 4 matrix}
b. W = {(a, b, c) : b = 2c + 1 − a}
c. W = {X : AX = 0 where A is a 4 × 3 matrix}
d. W = {(a, b, c) : a, b, c are even integers}
6. Show that the set S = {(2, 13, 2), (1, 6, 1), (1, 7, 3)} spans R3 . Express (−3, 2, −9) in
R3 as a linear combination of vectors in S. Is this linear combination unique,explain.
7. Let S = {(−1, 4, 2, 3), (2, −8, −4, 5)}. Enlarge this set to form a basis for R4 .
8. Let S = {(1, −2, 3, −5), (0, −1, 2, −3)}. Enlarge this set to form a basis for R4 .
9. Find the parametric equation of the line of intersection of the planes 2x + 3y − z =
3, x + 7y + 9z = 6.
10. For which real values of λ do the following vectors form a linearly dependent set in
R3 .
v1 = (λ , 1, 2), v2 = (1, λ , 2), v3 = (1, 2, λ )

11. Determine whether the following polynomials span P2 .

p1 = 1 + x + 2x2 , p2 = 3 + x, p3 = 5 − x + 4x2 , p4 = −2 − 2x + 2x2

12. Find the coordinate vector for v relative to the basis S = {v1 , v2 , v3 } for R3 .
a. v = (2, −1, 3); v1 = (1, 0, 0), v2 = (2, 2, 0), v3 = (3, 3, 3)
b. v = (5, −12, 3); v1 = (1, 2, 3), v2 = (−4, 5, 6), v3 = (7, −8, 9)
13. Find a basis for the subspace of R4 spanned by the given vectors.
a. (2, 4, −2, 3), (−2, −2, 2, −4), (1, 3, −1, 1)
b. (−1, 1, −2, 0), (3, 3, 6, 0), (9, 0, 0, 3)
c. (1, 1, 0, 0), (0, 0, 1, 1), (−2, 0, 2, 2), (0, −3, 0, 3).
 
1 3 1 −2 −3
1 4 3 −1 −4
14. Find a basis for the row space and column space of the matrix,  
2 3 −4 −7 −3
3 8 1 −7 −8

You might also like