無年度 - [非官方正解]主題課程_向量空間：線性獨立和線性相依#108250

科目：主題課程專用 | 選擇題數：2 | 申論題數：3

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所屬科目：主題課程專用

選擇題 (2)

 (10%)There are six vectors shown below. Please determine the right description (A) are linear independent (B) are linear independent (C) are linear independent (D) are linear independent 
 (5%) Which of the following lists of vectors in R3 are linearly dependent? (A)u=(1,2,5),t(2,5,1),u元(5,2,1) (B)u=(1,2,5), =(1,5,6), w=(0, 0, 0) (C)น=(2,-4,8),=(3 ,-6, 12),w=(1,2,6) (D)u-(1,2,3),=(4,5,6),w=(1,4,6),x-(3,4,6)

申論題 (3)

 (5 points) Defining the a1, a2,...，an  being n column vectors in, R2 and let A = [a1 a2 ... an],
(1)the vectors a1，a2，...， an will be linearly independent if and only if A is nonsingular. 
(2) the detcrminate of A is nonzero.
 (3) the vectors a1, a2,  an will be linearly independent if and only if the null space N(A) ={0}.
(4) the rank of A plus the nullity of A equals n.
(5) the rank of ATequals the rank of A.
(5 points) A nxn matrix A is said to be nonsingular, if
 (1) A is non-invertible.
 (2) there exists a multiplicative inverse matix B such that A is row cquivalent to B.
 (3) the determinate of A is nonzero.
 (4) Ax=O where x is nonzero vector.
 (5) A is row equivalent to L
(15%) Let x1, x2, and x3 be linearly independent vectors in Rnand let
 y1=x1 + x2, y2 =2x3-2x1,y3 =3x2+ 3x3
Are y1 y2, and y3 linearly independent? If it is true, please explain your reason; otherwise, please show that there are nontrivial linear combinations of y1 ，y2，y3 that equal 0.