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A. Dimension, Rank, Basis, Four Fundamental Subspace
B. Orthogonality, Projection, Component, Eigenvectors, and Eigenvalues
1. Find the rank of the matrix
a) π΄π΄ = οΏ½
1 2 3
2 3 4
3 5 7
οΏ½
b) B = οΏ½
0 1 2 1
1 2 3 2
3 1 1 3
οΏ½
2. Let ππ be a subset of R4 consisting of vectors that are perpendicular to vectors ππ, ππ, ππππππ ππ
where ππ = < 1, 0, 1, 0 >, ππ = < 1, 1, 0, 0 >, ππ =< 0, 1, −1, 0 >,
Namely, ππ = {π₯π₯ ∈ π
π
4|πππππ₯π₯ = 0, πππππ₯π₯ = 0, ππππππ πΆπΆπππ₯π₯ = 0}
a. Prove that V is a subspace of π
π
4
b. Find a basis for V
c. Determine the Dimension of V
Solution Hint:
a) Observe that the conditions πππππ₯π₯ = 0, πππππ₯π₯ = 0, ππππππ πππππ₯π₯ = 0, combining π΄π΄π΄π΄ = 0
Where, A = οΏ½
1 0 1 0
1 1 0 0
0 1 −1 0
οΏ½. Note that the rows of the matrix A are ππππ, ππππ, ππππππ ππππ. It follows
that the subset V is in the null space ππ(π΄π΄) of the matrix π΄π΄. Being the null space ππ = ππ(π΄π΄), is
a subspace of π
π
4.
b) To find a basis, we determine the solutions of π΄π΄π΄π΄ = 0
Applying elementary row operations to the augmented matrix, we see that,
οΏ½
1 0 1 0 0
1 1 0 0 0
0 1 −1 0 0
οΏ½ βΆ οΏ½
1 0 1 0 0
0 1 −1 0 0
0 1 −1 0 0
οΏ½ (π
π
2 − π
π
3) βΆ οΏ½
1 0 1 0 0
0 1 −1 0 0
0 0 0 0 0
οΏ½
Then, Determine the general solution and determine the basis and you will have it.
3. Determine which of the following is a subspace of π
π
3.
a) π₯π₯ + 2π¦π¦ − 3π§π§ = 4
b) π₯π₯−1
2 = π¦π¦+2
3 = π§π§
4
c) π₯π₯ + π¦π¦ + π§π§ = 0 and π₯π₯ − π¦π¦ + π§π§ = 1
d) π₯π₯ = −π§π§ and π₯π₯ = π§π§
e) π₯π₯2 + π¦π¦2 = π§π§
f) π₯π₯
2 = π¦π¦−3
5
4. Suppose ππππππππ(π
π
0) = π΄π΄ π€π€βππππππ π
π
0 = οΏ½
1 3 5 0 7
2 6 10 1 16
3 9 15 1 23
οΏ½. Show that –
a) The row space has dimension 2, matching the rank
b) The column space of π
π
0 has also dimension ππ = 2
c) The null space of π
π
0 has dimension 3
d) The null space of π
π
0
ππ, which can also be called the left null space of π
π
0; has dimension 1.
5. Find a basis for each of the four fundamental subspaces associated with the matrix.
A= οΏ½
1 2 0 1
0 1 1 0
1 2 0 1
οΏ½
6. Let A be a real 7x3 matrix such that the null space is spanned by the vectors
οΏ½
1
2
0
οΏ½,οΏ½
2
1
0
οΏ½ ππππππ οΏ½
1
−1
0
οΏ½. Find the rank of the matrix A.
7. Let V be a subset of the vector space π
π
ππ consisting only of the zero vector of π
π
ππ, Namely ππ =
{0}. Then prove that V is a subspace of π
π
ππ.
8. Let π΄π΄ = οΏ½
4 1
3 2
οΏ½and consider the following subset V of the 2-dimensional vector space π
π
2,
Namely ππ = {π₯π₯ ∈ π
π
2|π΄π΄π΄π΄ = 5π₯π₯}
a) Prove that the subset V is a subspace of π
π
2
b) Find a basis for V and determine the dimension of ππ
9. The smallest subspace of π
π
3 containing the vectors (2, -3, -3) and (0, 3, 2) is the plane whose
equation is ππππ + ππππ + 6π§π§ = 0. Determine the value of ππ, ππ.
10. Determine The matrix representation of the orthogonal projection operator taking π
π
3 onto
the plane π₯π₯ + π¦π¦ + π§π§ = 0.
11. Let π’π’ = (8, √3, √7, −1, 1) and π’π’ = (1, −1, 0, 2, √3). If the orthogonal projection of u onto v
is ππ
ππ
π£π£, then determine a and b.
12. Find the point ππ in π
π
3 on the ray connecting the origin to the point (2, 4, 8) which is closest
to the point (1, 1, 1).
13. Find the eigenvalues and eigenvectors of the following matrix A.
π΄π΄ = οΏ½
1 −1 0
−1 2 −1
0 −1 1
οΏ½
Show that these eigenvectors are perpendicular. [Hint: It will always be perpendicular when
A is symmetric]
14. Suppose you want a vector to rotate about 90 Degree anti-clockwise. Determine the
transformation matrix that should operate on that vector to produce such result? Determine
for 180, and 270 degrees too.
15. Find the rank and the four eigenvalues of A, where π΄π΄ = οΏ½
1 0 1 0
0 1 0 1
1 0 1 0
0 1 0 1
οΏ½
16. [Page 201, Worked Example 4.1A, Introduction to Linear Algebra (4th Edition), Gilbert
Strang]
Suppose S is a six-dimensional subspace of nine-dimensional space π
π
9.
a. What are the possible dimensions of subspaces orthogonal to ππ?
b. What are the possible dimensions of the orthogonal complement ππ^ of S?
c. What is the smallest possible size of a matrix A that has row space S?
d. What is the shape of its null space matrix N?
17. (Bonus Problem)
Find all eigenvalues and eigenvectors of the matrix π΄π΄,
π€π€βππππππ π΄π΄ =
β£
β’
β’
β’
β’
β‘
10001 3 5 7 9 11
1 10003 5 7 9 11
1 3 10005 7 9 11
1 3 5 10007 9 11
1 3 5 7 10009 11
1 3 5 7 9 10011β¦
β₯
β₯
β₯
β₯
β€
Solution Hint: Let π΅π΅ = π΄π΄ − 10000πΌπΌ, where I is the 6 ∗ 6 identity matrix. That is, we have,
π΅π΅ =
β£
β’
β’
β’
β’
β‘
1 3 5 7 9 11
1 3 5 7 9 11
1 3 5 7 9 11
1 3 5 7 9 11
1 3 5 7 9 11
1 3 5 7 9 11β¦
β₯
β₯
β₯
β₯
β€
, since all row are same, π΅π΅ is singular and hence ππ = 0 is an
eigenvalue of B. With elementary row operation, we find π΅π΅ βΆ
β£
β’
β’
β’
β’
β‘
1 3 5 7 9 11
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 β¦
β₯
β₯
β₯
β₯
β€
By inspection, we see that π΅π΅π΅π΅ = 36π£π£, where π£π£ =< 1, 1, 1, 1, 1, 1 >. Thus it yields that ππ = 36 is
the eigenvalue of π΅π΅ ππππππ π£π£ is the corresponding eigenvector.