Answer:
(A) Set A is linearly independent and spans
. Set is a basis for
.
Step-by-Step Explanation
<u>Definition (Linear Independence)</u>
A set of vectors is said to be linearly independent if at least one of the vectors can be written as a linear combination of the others. The identity matrix is linearly independent.
<u>Definition (Span of a Set of Vectors)</u>
The Span of a set of vectors is the set of all linear combinations of the vectors.
<u>Definition (A Basis of a Subspace).</u>
A subset B of a vector space V is called a basis if: (1)B is linearly independent, and; (2) B is a spanning set of V.
Given the set of vectors
, we are to decide which of the given statements is true:
In Matrix
, the circled numbers are the pivots. There are 3 pivots in this case. By the theorem that The Row Rank=Column Rank of a Matrix, the column rank of A is 3. Thus there are 3 linearly independent columns of A and one linearly dependent column.
has a dimension of 3, thus any 3 linearly independent vectors will span it. We conclude thus that the columns of A spans
.
Therefore Set A is linearly independent and spans
. Thus it is basis for
.
I got 6 if that helps i dont know if thats right or not
Hey there. To find the answer solve 8^3=512.
Count the rows and columns of each to get C as your answer.
Y-4X =3 ; 2x-3y=21
Y= 3+ 4x
2x -3( 3+4x)= 21
2x- 9- 12x= 21
-10x -9= 21
-10x -9+9= 21+9
-10x = 30
X= 30/-10= -3
Y= 3+4x = 3+ 4* (-3)= 3-12= -9