선형대수학 TF 문제

내가 선형대수학을 공부하며 마주친 TF문제를 정리해보고자 합니다

 

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1) The pivot positions in a matrix depend on whether row interchanges are used

in the row reduction process.

- FALSE

 

2) Whenever a system has free variables, the solution set contains many solutions.

- FALSE (첨가행렬 [(1 1 0),(0 0 1)]

일때를 상상해보자. 이미 두 번째 COLUMN에 FREE VAR.가 있는데 해가 없는 상황이다.)

 

3) A general solution of a system is an explicit description of all solutions of the

system.

- TRUE

 

4) The equation Ax = b is consistent if the augmented matrix [A b] has a pivot

position in every row.

- FALSE

 

5) If the equation Ax = b is inconsistent, then b is not in the set spanned by the

columns of A.

- TRUE

 

6) If A is an m × n matrix whose columns do not span Rm, then the equation Ax = b is inconsistent for some b in Rm.

- TRUE

 

7) If A is a 3 × 5 matrix and T is a transformation defined by T(x) = Ax, then

the domain of T is R3.

- FALSE (DOMAIN : 정의역 // 여기서 x는 5차원 - column개수)

 

8) A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2)

for all v1 and v2 in the domain of T and for all scalars c1 and c2.

- TRUE(벡터의 덧셈과 스칼라배가 가능 – 이는 선형사상의 특징)

 

9) A function f : R → R defined by f(x) = mx + b is a linear transformation if

and only if b = 0.

- TRUE (b가 있다면 f(x) = mx+b , f(y) = my+b // f(x+y) = m(x+y)+2b가 되어버린다.

 

10) If the columns of an n × n-matrix A span Rn, then the columns are linearly

independent.

- TRUE (n차원을 생성하기 위해선 n개의 basis, 즉 independent vectors가 필요함.

 

**11) If the equation Ax = b has at least one solution for each b in Rn, then the solution is unique for each b.

- True; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in set of real numbers ℝn​, then matrix A is invertible. If A is​ invertible, then according to the invertible matrix theorem the solution is unique for each b.

 

12) If there is a b in Rn such that the equation Ax = b is inconsistent, then the

transformation x |→ Ax is not one-to-one.

- TRUE(according to the Invertible Matrix Theorem if there is a b in set of real numbers ℝn such that the equation Ax=b is​ inconsistent, then equation Ax=b does not have at least one solution for each b in set of real numbers ℝn and this makes A not invertible.)

 

13) A subspace of Rn is any set H such that (i) the zero vector is in H, (ii) u, v,

and u + v are in H, and (iii) c is a scalar and cu is in H.

- TRUE

 

14) If v1, . . . , vp are in Rn, then Span{v1, . . . , vp} is the same as the column space of the matrix [v1 · · · vp].

- TRUE

 

**15) The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of Rm.

- FALSE(솔루션인 벡터는 should be a subspace of Rn // Rm이 아니라 Rn이면 true일수도!)

 

16) The columns of an invertible n × n matrix form a basis for Rn.

- TRUE (LI)

 

17) Row operations do not affect linear dependence relations among the columns

of a matrix.

- TRUE

 

(a) For two n × n matrices A and B, we have (det A)(det B) = det AB.

-TRUE

 

(b) If Rn has a basis of eigenvectors of A, then A is diagonalizable.

- TRUE

 

(c) If A is diagonalizable, then A is invertible.

- False - Invertibility doesn't affect diagonalizability. A matrix is invertible if 0 is not an eigenvalue. A diagonalizable matrix may or may not have 0 as an eigenvalue.

 

(d) If A is invertible, then A is diagonalizable.

- False - Invertibility and diagonalizability do not affect each other and are two completely different concepts.

 

(e) If AP = P D with D diagonal, then the nonzero columns of P must be eigenvectors

- True - Each column of PD is a column of P times A and is equal to the corresponding entry in D times the vector P. As long as the column is nonzero, the equation AP = PD is valid.

 

(f) If A is similar to B, then A^2 is similar to B^2

– TRUE

 

(a) Not every orthogonal set in Rn is linearly independent.

- TRUE – 영벡터가 있다면 dependent set

 

(b) A matrix with orthonormal columns is an orthogonal matrix.

- FALSE – MATRIX가 SQUARE인 조건도 있어야한다.

 

(c) If y is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.

- TRUE

 

(d) The orthogonal projection yˆ of y onto a subspace W can depend on the orthogonal basis for W used to compute yˆ.

- FALSE

 

(e) If an n × p matrix U has orthonormal columns, then UUT x = x for all x in Rn

- FALSE – U 가SQUARE 일때만 TRUE

 

(f) If y = z1 + z2, where z1 is in a subspace W and z2 is in W⊥, then z1 must be the orthogonal projection of y onto W.

- TRUE

 

(g) The Gram-Schmidt process produces from a linearly independent set {x1, . . . , xp} an orthogonal set {v1, . . . , vp} with the property that for each k, the vectors v1, . . . , vk span the same subspace as that spanned by x1, . . . , xk.

- TRUE

 

(h) If x is not in a subspace W, then x − projW x is not zero.

- TRUE

 

(a) The general least-squares problem is to find an x that makes Ax as close as

possible to b. - TRUE

 

(b) A least-squares solution of Ax = b is a vector xˆ that satisfies Axˆ = bˆ, where

bˆ is the orthogonal projection of b onto Col(A).

- TRUE

 

(c) Any solution of AT Ax = ATb is a least-squares solution of Ax = b.

- TRUE

 

(d) The least-squares solution of Ax = b is the point in the column space of A

closest to b. - FALSE , x hat에 관한 설명이다.

 

(a) If AT = A and if vectors u and v satisfy Au = 3u and Av = 4v, then u•v = 0.

- TRUE

 

(b) An n × n symmetric matrix has n distinct real eigenvalues.

- FALSE eigenvector는 n개 있을거다.

 

(c) If B = P D P^T, where P^T = P^−1 and D is a diagonal matrix, then B is a symmetric matrix.

- TRUE

 

(a) A Positive definite quadratic form Q satisfies Q(x) > 0 for all x ∈ Rn.

false – x=0일 때 제외해야됨

 

(b) The expression ||x||2 is not a quadratic form.

- TRUE

 

(c) If A is a 2 × 2 symmetric matrix, then the set of x such that xT Ax = c (for a constant c) corresponds to either a circle, an ellipse, or a hyperbola.

- FALSE (0벡터, 즉 EMPTY일수도)

 

(d) An indefinite quadratic form is neither positive semidefinite nor negative semidefinite.

- FALSE

 

 

틀린설명이나 답은 댓글러 알려주시면 감사하겠습니다

 

REFERENCE

• David Lay - Linear Algebra and its application을 독학하며 정리한 내용입니다.