Math212 Take Home Midterm 1
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Take Home Midterm 1
Due February 26 2023
1 Section 1B - Vector Spaces
Problem 1 Write down the definition of vector space over a field F.
Problem 2 Let P[x] be the collection of all polynomials in the variable x with real coefficients. That is, for each p ∈ P[x] there is some n ∈ N ∪ {0} and real numbers a0 , ...,an ∈ R so that p = a0 + a1 x + ... + an xn . For example 2 + πx + 9.7x2 − 5x9 is a polynomial in P[x]. Define scalar multiplication by c(a0 + a1 x + ... + an xn ) = ca0 + ca1 x + ... + can xn where c ∈ R. If n ≥ m we define vector addition by (a0 + a1 x + ... + an xn ) + (b0 + b1 x + ... + bm xm ) = ((a0 + b0 ) + (a1 + b1 )x + ... + (am + bm )xm + am+1xm+1 + ... + an xn (this is the standard addition of polynomials). Prove or disprove that P[x] is a vector space.
Problem 3 Let V = R / {0} be the set of all real numbers except 0. Define the following vector addition and scalar multiplication. Define v1 + v2 = v1 v2 where v1 ,v2 ∈ V and define cv = vc where c ∈ F = R and v ∈ V . Show that V is a vector space and find the vector. Note that the number 0 is NOT in V so
the zero vector is NOT the number 0.
Problem 4 List 5 examples of vector spaces other than ones given here. No proof is necessary, but you should clearly state what vector addition and scalar multiplication are.
2 Section 1C - Subspaces
Problem 1 Write down the definition of a subspace of a vector space.
Problem 2 Is R a subspace of R2 ? Why or why not?
Problem 3 For which c,λ ∈ R is Vλ = {p ∈ P[x] : p(λ) = c} a subspace of P[x]?
Problem 4 Let V be a vector space over R. Let W1 ,W2 be two subspaces. Prove or disprove that W1 ∩ W2 is a subspace of V .
3 Section 2A - Span and Linear Independence
Problem 1 Write down the definition of what it means for vectors v1 , ...,vn ∈ V to span V . Write down the definition of what it means for vectors v1 , ...,vn ∈ V to be linearly independent.
Problem 2 Consider the set of vectors x,x + 3x2 ,x3 ,x2 ∈ P[x]. Is this set linearly independent? Does this set span P[x]?
Problem 3 Let S1 ,S2 ⊂ V be two sets of vectors which each span V , show that S1 n S2 span V .
Problem 4 Let S1 ,S2 ⊂ V be two sets of vectors which are each linearly independent, show that S1 ∩ S2 is linearly independent.
4 Section 2B - Bases
Problem 1 Write the definition of basis of vector space.
Problem 2 Give a basis for P[x].
Problem 3 Let B1 ,B2 ⊂ V be two bases of vector space V with B1 B2 . Is it true that B1 ∩ B2 is also a basis for V? Why or why not? Is it true that B1 n B2 is also a basis for V? Why or why not?
Problem 4 Suppose that v1 ,v2 ,v3 ,v4 is a basis for vector space V . Is it true that v1 + 2v2 ,v2 + 2v3 ,v4 is also a basis for V?
5 Section 2C, 3A and 3B - Dimension, Linear Transformations, Null Spaces and Ranges
Problem 1 Write down the definition of dimension of vector space.
Problem 2 What is the dimension of P[x]≤12 = {p ∈ P[x] : degreep ≤ 12}?
Problem 3 Let T : V → W be a function. Write what it means for T to be linear.
Problem 4 Let V,W be finite dimensional vector spaces. If T : V → W is linear, injective and surjective show that the dimension of V is the dimension of W . You may only use results from Section 3B and before.
2023-02-18