线性代数作业代写 MTH 342-020 OSU代写 数学作业代写代做

MTH 342-020 OSU Spring 2022 Written Homework #1

线性代数作业代写 Turn this in by making a pdf scan of your work and submitting it to Gradescope by 11:59 pm on the due date. This needs to submitted as a variable

Turn this in by making a pdf scan of your work and submitting it to Gradescope by 11:59 pm on the due date. This needs to submitted as a variable length submission.

How to write up homework. Show your work and explain it. In general, just writing down an answer without showing the steps to arriving at the answer is not sufficient for credit.

Read pages 12 – 16 in LADW. (This is in addition to the previous reading: pp. 1- 11 in LADW. pp. 36 – 42 of the UC Davis text.)

1. 线性代数作业代写

An n × n matrix is called symmetric if A^T = A. Let

V = {A ∈ M_n×n | A is symmetric}.

(a) Show that V is a vector space, by showing that V is a subspace of Mn×n. Use Lemma 4.3.2 on page 40 in the UC Davis text: show that

(i) The additive identity belongs to V : 0_n×n ∈ V ,

(ii) V is closed under addition,

(iii) V is closed under scalar multiplication.

(This method of showing V is a vector space is less tedious than verifying that V satisfies all of the axioms of a vector space.)

(b) Determine a basis for V . Hint: Let B_ij denote the matrix whose (i, j)-and (j, i)-entries are 1, with 0’s elsewhere.

(c) What is the dimension of V ? (The dimension of a vector space V is the number of elements in a basis for V .)

2. 线性代数作业代写

Let L be the following subspace of M_n×n:

L = {A ∈ M_n×n | A is lower triangular}.

Hint: Let E_ij denote the matrix whose (i, j)-entry is 1, with 0’s elsewhere.

(a) Determine a basis for L.

(b) What is the dimension of L?

(a) T : R² → R³ defined by T(x, y)^T = (x + 2y, 2x − 5y, 7y)^T .

(b) T : C³ → C² given by T(x, y, z)^T = (x + iy, x − iz)^T .

(c) T : P₄ → P₂ given by T f(x) = f’’(x). Use the standard basis {1, x, x² , …, xⁿ} for P_n. The matrix will have size 3 × 5.

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