課程名稱︰線性代數 課程性質︰必修 課程教師︰蘇柏青 開課學院：管理學院 開課系所︰工管系科管組 考試日期（年月日）︰104/10/30 考試時限（分鐘）：50 min 試題 : Linear Algebra Quiz #1 1. Properties of Matrices(80%) - - | 1 1 3 0 1 | | 1 1 3 0 1 | Consider the matrix A = | 0 0 0 1 -1 | | 0 1 1 0 0 | - - (a) (4%) How many columns does A have? How many rows does A have? (b) (10%)Find the reduced row echelon form R for the matrix A using Gaussian Elimination. (Keep track of each elementary row operation you performed to obtain R for future use) (c) (8%) For each elementary roe operation you did in (b), write the correspon ding elementary matrix. (d) (5%) Find an invertible matrix P such that R = PA (Hint:Use results in (b)) (e) (10%)Write P as a product of several elementary matrices. (f) (6%) What is the rank of A? What is the nullity of A? (g) (3%) Which columns of R are pivot columns? (h) (9%) Let ai denote the ith column of A. Is {a1,a2} linearly independently? Is{a1,a2,a3} linearly independent? Is {a1,a2,a4} linearly independent?Why? T (i) (7%) Is the vector b =[ 1 1 2 3] in the span of all columns of A? Δ (j) (6%) Let T be a linear transformation induced by matrix A, i.e,T(x) = Ax. What is the domain of T? What is the codomain of T? (k) (6%) What is the range of T? Is T an onto transformation? (l) (6%) Find the null space of T (i.e., the set of all v in the domain suth that T(v) = 0). Is T a one-to-one transformation? 2. Proof of theorems(20%) (a) (10%) Let S={u1,u2,…,uk} be a finite subset of R^n. Prove that if u and v are both in Span S, then u+cv is also in Span S for any scalar c. (b) (10%) Let u and v be distinct vectors in R^n. Prove that the set {u,v} is linearly independent if and only if the set {u+v,u} is linearly independent. --

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