無年度 - 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

複選題
(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,...，a_n being n column vectors in, R² and let A = [a₁ a₂ ... an],

(1)the vectors a1，a2，...， a_n will be linearly independent if and only if A is nonsingular.

(2) the detcrminate of A is nonzero.

(3) the vectors a₁, a₂, a_n 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 A^Tequals 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.