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MAT A22

Homework # 4 –

Winter 2024

Homework Guidelines

This homework was released on Fri. Feb. 2nd 14:00 (EST). It is due on Fri. Feb. 10th 17:00 (EST).

We encourage you to talk to your TAs during tutorial, attend office hours, and ask professors for help with this assignment. You may use the textbook without citing it as a reference, however all other books and internet sources must be cited. Please submit your original work via Crowdmark.

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Questions 1-5 could be graded for this homework assignment. Question 6 is meant to be a fun exercise.

Readings

❼ ➜1.6 Bases and DimensionMAT A22 Winter 2024

Problems

Q1. (a) The vectors {(3, 2, 1),(2, 1, 0)} are linearly independent in R3.

Find a vector v such that {(3, 2, 1),(2, 1, 0), v} is a basis of R3.

Use the elimination algorithm to prove v ∈ span({(3, 2, 1),(2, 1, 0)}).

(b) The vectors {(1, 2, 3),(1, 1, 2),(0, 1, 1),(1, 0, 0)} form a span-ning set in R 3 . Use the elimination algorithm to determine which vector can be removed to create a basis.

Q2. Prove: If W ⊆ V is a infinite dimensional subspace of a vector space V then V is infinite dimensional.

Q3. Suppose that V is finite dimensional. Let W1 and W2 be subspaces of V with W1 ∩ W2 = {0} and W1 + W2 = W3

Prove: dim(W3) = dim(W1) + dim(W2).

Q4. Find the dimension of the following subspaces.

(a) W are the trace zero real 2 × 2 matrices.

(b) W are the skew symmetric real 2 × 2 matrices.

(c) Polynomials of degree at most n.

Q5. Suppose we have two different subsets S1 and S2 of a finite vector space V , where |S1| < |S2|. If the dim(span(S1)) = dim(span(S2)).

What can we say about the dependency of S2?

Q6. Define a set W that is a subspace of C 3 with dim(W) = 2 such that the basis for W must contain an entry the is not real.