Linear Algebra 2270
Homework 11
a preparation for the last quiz on 08/05/2015
Problems:
1. Perform Gram-Schmidt orthogonalization of the following vectors:
⎡1⎤
⎡−1⎤
⎡−2⎤
⎡2⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
v1 = ⎢ 0 ⎥ , v2 = ⎢ 1 ⎥ , v3 = ⎢ 1 ⎥ , v4 = ⎢0⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢−1⎥
⎢3⎥
⎢4⎥
⎢4⎥
⎣ ⎦
⎣ ⎦
⎣ ⎦
⎣ ⎦
And using it find matrices Q̃, R̃ such that matrix
⎡∣
∣
∣
∣ ⎤⎥
⎢
⎥
⎢
A = ⎢v1 v2 v3 v4 ⎥ = Q̃R̃
⎥
⎢
⎢∣
∣
∣
∣ ⎥⎦
⎣
Use this factorization of A to find the solution set of the linear system:
Ax = 0
2. Perform Gram-Schmidt orthonormalization of the following vectors:
⎡ √2 ⎤
⎡ 1 √2⎤
⎡−√2⎤
⎡√2⎤
⎢
⎥
⎢2 ⎥
⎢
⎥
⎢ ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ⎥
0 ⎥ , v2 = ⎢ √
0 ⎥ , v3 = ⎢ √
0 ⎥ , v4 = ⎢√1 ⎥
v1 = ⎢ √
⎢
⎥
⎢3 ⎥
⎢
⎥
⎢ ⎥
⎢− 2⎥
⎢ 2⎥
⎢2 2⎥
⎢ 2⎥
⎣
⎦
⎣2 ⎦
⎣
⎦
⎣ ⎦
And find QR decomposition of matrix
⎡∣
∣
∣
∣ ⎤⎥
⎢
⎢
⎥
A = ⎢v1 v2 v3 v4 ⎥
⎢
⎥
⎢∣
∣
∣
∣ ⎥⎦
⎣
1
3. (a) Consider space R2 with the dot product and a set W = {[ ]}. Find W ⊥ and (W ⊥ )⊥
−1
1
]). Find W ⊥ and (W ⊥ )⊥
−1
⎡ 1 ⎤ ⎡0⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
(c) Consider space R3 with the dot product and a set W = {⎢−1⎥ , ⎢0⎥}. Find W ⊥ and (W ⊥ )⊥
⎢ ⎥ ⎢ ⎥
⎢ 0 ⎥ ⎢1⎥
⎣ ⎦ ⎣ ⎦
(d) Let V be any inner product space. Let W be any subset of V , W ⊂ V . Prove that W ⊥ is a
subspace.
(b) Consider space R2 with the dot product and a set W = span([
(e) Let V be a finite dimensional inner product space. Let W be any subspace of V , W ≺ V .
Prove that W = (W ⊥ )⊥ .
Hint: Start by considering an orthonormal basis of W , enrich it to an orthonormal basis of V ,
show that the additional vectors form a basis of W ⊥ .
4. (a) Find a determinant of A by performing row or column operations (or both). [In additional
notes posted on the website, we prove that one can perform row operations. The use of column
operations have been justified in the lecture.]
⎡2 2 2 ⎤
⎢
⎥
⎢
⎥
A = ⎢1 0 1 ⎥
⎢
⎥
⎢2 −2 1⎥
⎣
⎦
1
(b) Find a determinant of A by performing expansions according to rows or columns. For 2 × 2
matrices you may use a formula given in the lecture.
⎡2
⎢
⎢1
⎢
A=⎢
⎢2
⎢
⎢3
⎣
2 2
0 1
0 1
0 −1
1⎤⎥
2⎥⎥
⎥
0⎥⎥
1⎥⎦
Hint: It might be a good idea to start with an expansion according to the second column.
(c) Using all available methods (expansion according to a row or a column or row and column
operations), find the determinant of the matrix:
⎡2
⎢
⎢2
⎢
A=⎢
⎢2
⎢
⎢3
⎣
3 2 1⎤⎥
3 4 1⎥⎥
⎥
1 −1 0⎥⎥
0 −1 1⎥⎦
Hint: It might be a good idea to start with operations: r2 + = (−1)r1 , r2 = 21 r2 and then an
expansion according to one of the rows.
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