複選題

Which of the following are linear cambinations of and?
(A)
(B)
(C)
(D)

Suppoe that T: R2 → R2 is a linet tansformatian such that T and , = ?(A)(B)(C)(D)

複選題 Which of the following is true?(A) Every function from , T : , has a standard matrix A such thatT(x) = Ax for all x ∈ .(B) The matrix transformation induced by an m X n matrix A (i.e., ) is a lineartransformation.(C) The image of the zero vector under any linear transformation is the zero vector.(D) A function T : is uniquely determined by the images of the standard vectorsin its domain.(E) If f is a linear transformation and f(u) = f(v), then u = v.

複選題Which of the following is true?(A) A linear transformation with codomain is onto if and only if the rank of its standardmatrix is m.(B) A linear transformation is one-to-one if and only if its null space consists only of the zerovector.(C) A linear transformation is onto if and only if the columns of its standard matrix form agenerating set for its range.(D) If the composition UT of two linear transformations T : and U : isdefined, then m = p must be true and the composition UT is also a linear transformation.(E) For every invertible linear transformation T, the function is also a linear transfor-mation.

複選題Which of the following is a linear operator?(A) T : R→R where T(x) = 40 + 3 for any I E R.(B) T:T2→R2 where for any vector x∈ R2?.(C) T : P →P where T(f(x)) = f(x)(x2 + 1) for any polynomial f(x) ∈ P.(D) T : where T(f(x)) = f'(x)+ f(x) for any differentiable functionf(x) ∈ C(R).(E) None of the above.

複選題Which of the following are correct?(A) The system defined by F(x,y)=(x2, x) is linear.(B) The system defined by F(x,y)=(dx/dt, x) is linear.(C) A and B are m x n matrices. If Aw = Bw for all w in , then A = B.(D) The columns of matrix A contains zero vector. If Ax=b have solution, it will have Infinite solutions.(E) The zero vector of R" is within the span of any finite subset of .

複選題Which of the following transformations are not linear?(A) S: the map in R3 which rotates points about the x1-axis by an angle π/2.(B) (C) (D) T(ax2+bx+c)-(a+b)x+(b+c)(E)

Let A be an m x n matrix whose null space has dimension k. Which conclusion is correct? (A) The dimension of Null(AT) is k. (B) The dimension of row space of A is m-k. (C) The dimension of column space of A is m-k. (D) The dimension of row space of A'is n-k. (E) The dimension of column space of A is n-k.

Let A =xyT, where t and y are two nonzero vectors of Rn, n ＞ 1. Which of the following statements is/are true? (A) rank( A ) = 1 and the range space of A is Spanf{y}. (B) rullity( A ) = 2 and the null space of A is Spanfe,{x，y}. (C) Trace( A ) = 1 and det( A ) =0 (D) A is always diagonalizable. (E) None of the above.

.(5%) Determine the rank of the matrix in the following.  (A)0 (B)1 (C)2 (D)3 (E) 4

11. (10%) Let Q be an n✖ n matrix. Then which of the following set is not a subspace? (A) Col O (B) Null Q (C) rank Q (D) Row Q (E) None of the above

無年度 - 主題課程_向量空間：秩(rank)、零度(nullity)、維度#108952

無年度 - 主題課程_向量空間：基底和維度#108872

無年度 - 主題課程_理工學院機率論：古典機率論#108850

無年度 - 主題課程_向量空間：線性獨立和線性相依#108250

無年度 - 主題課程_理工學院機率論：機率分配（離散型、連續型）#107979

無年度 - 主題課程_向量空間：向量空間和子空間#107951

無年度 - 主題課程_理工學院機率論：多元隨機變數#107950

無年度 - 主題課程_理工學院機率論：單一隨機變數#107935

無年度 - 主題課程_行列式和線性方程式：線性方程式#107934

無年度 - 主題課程_線性映射：基底變換和座標變換#107933