CSCI 2820 Fall 2020
HW 4 /100
Due: Thursday, October 1, at midnight, via
Gradescope
September 23, 2020
1. [Linear Independence of block vectors] (/20) Consider the block vectors
−→ci = (−→ai ,−→bi ), i = 1, · · · , k
where the −→ai are n-vectors and the −→bi are m-vectors.
1. Suppose the −→ai are linearly independent. Can we conclude that the−→ci are linearly independent? please explain your answer.
2. Suppose that the −→ai are linearly dependent. Can we conclude that the−→ci are also linearly dependent? Please explain your answer.
2. [Orthogonalizing vectors?] (/15)
Suppose that −→a ,−→b are arbitrary n-vectors. Show that we can always
find a scalar γ such that (−→a − γ−→b ) ⊥ −→b and γ is unique if −→b 6= −→0 .
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3. [Gram-Schmidt] (/15)
Consider the list of n n-vectors−→a1 = (1, 0, 0, · · · , 0),−→a2 = (1, 1, 0, · · · , 0), · · · ,−→an =
(1, 1, · · · , 1).Describe what happens if we run Gram-Schmidt on the list of−→ai , i.e. what are the vectors −→qi we obtain? Is the collection of −→ai a basis?
4. [Gram-Schmidt twice] (/15)
We run the Gram-Schmidt algorithm on a given set of vectors −→a1,, · · · ,−→ak
(we assume this is successful), which gives the vectors −→q1,, · · · ,−→qk . Then we
run the Gram-Schmidt algorithm on the vectors −→q1,, · · · ,−→qk which produces
the vectors −→z1,, · · · ,−→zk . What can you say about −→z1,, · · · ,−→zk?
5. [Early termination of Gram-Schmidt] (/15)
When the Gram-Schmidt algorithm is run on a particular list of 10 15-
vectors, it terminates in iteration 5 (−→q5 = −→0 ). Which of the following must
be necessarily true?
1. −→a2,−→a3,−→a4 are linearly independent.
2. −→a1,−→a2,−→a5 are linearly dependent.
3. −→a1,−→a2,−→a3,−→a4,−→a5 are linearly dependent.
4. −→a4 is nonzero.
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6. [Vector Spaces] (/20)
For each, list three elements and then show it is a vector space.
1. The set of linear polynomials P1 = {a0 + a1x |a0, a1 ∈ R} under the
usual polynomial addition and scalar multiplication operations.
2. The set of linear polynomials {a0 + a1x |a0 − 2a1 = 0}, under the
usual polynomial addition and scalar multiplication operations.
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