Question

(2 Points
If u=(1,0,0), v=(0.1.0) & w=(0,0,1)
then which of the following is/ are true?
1. V. u+vy are linearly independent vectors
U. V. w are linearly independent vectors
u is a linear combination of v and w
0.u+y.U+V+w; are linearly dependent vectors​

Answer 1

Given: If u=(1,0,0), v=(0.1.0) & w=(0,0,1)

To Find: Which of the following is/ are true?

              V. u+vy are linearly independent vectors

              U. V. w are linearly independent vectors  is a linear combination of        v and w

Solution:

Let's assume that the vectors  (+),(+)(u+v),(v+w)  and  (+)(u+w)  are not linearly independent.

⇒⇒  There exists scalers  ,a,b  and  ,c,  not all  0,0,  such that  (+)+(+)+(+)=0.a(u+v)+b(v+w)+c(u+w)=0.

⇒(+)+(+)+(+)=0.⇒(a+c)u+(a+b)v+(b+c)w=0.

However, since  ,u,v  and  w  are linearly independent the above equation would be valid if and only if  +=0,+=0a+c=0,a+b=0  and  +=0.b+c=0.

+=0⇒=−.a+c=0⇒c=−a.

+=0⇒=−.a+b=0⇒b=−a.

⇒+=−−=−2.⇒b+c=−a−a=−2a.

Then,  +=0b+c=0   ⇒−2=0⇒=0.⇒−2a=0⇒a=0.

⇒=−=0⇒b=−a=0  and  =−=0.c=−a=0.

But this contradicts our assumption that there exist scalers  ,a,b  and  ,c,  not all 00.

Therefore, the vectors  (+),(+)(u+v),(v+w)  and  (+)(u+w)  are also linearly independent.

Answer 2

They are linearly independent, the vectors (+), (+)(u+v) , (v+w), and (+)(u+w) are also independent of one another.

How do you know if a vector is linearly dependent?

• Two vectors are linearly dependent if and only if they are collinear, i.e., one is a scalar multiple of the other.
• Any set containing the zero vector is linearly dependent.
• If a subset of { v 1 , v 2 ,..., v k } is linearly dependent, then { v 1 , v 2 ,..., v k } is linearly dependent as well.

What is meant by linear dependence of vectors?

• The property of one set (as of matrices or vectors) having at least one linear combination of its elements equal to zero when the coefficients are taken from another given set and at least one of its coefficients is not equal to zero.

What is the difference between linearly dependent and linearly independent?

• In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector.
• If no such linear combination exists, then the vectors are said to be linearly independent.
• These concepts are central to the definition of dimension.

According to the question:

Given: If u=(1,0,0), v=(0.1.0) & w=(0,0,1).

The vectors (+),(+)(u+v),(v+w), and (+)(u+w) are not linearly independent, so let's suppose that.

There are scalers, a, b, and c, which are not all 0, 0, and which make (+)+(+)+(+)=0. a(u+v)+b(v+w)+c(u+w)=0.

= (+)+(+)+(+)=0.

= (a+c)u+(a+b)v+(b+c)w=0.

The preceding equation would only be true if and only if ,

+= 0,+ = 0a+c = 0,a+b = 0.

Because,u,v and w are linearly independent and + = 0.b+c = 0.

+=0.

= −a+c = 0.

= c =−a.

+= 0.

= −.a+b = 0.

b =−a.

+=−−=−2.

b+c =−a−a.

= −2a.

Then,  + = 0b+c = 0 .

−2=0 =0.

−2a = 0

a=0.

=−0

b=−a=0  and − = 0.c = −a = 0.

However, this goes against our presumption that there are scalers,a, b, and c, not just all 00.

Hence, they are linearly independent, the vectors (+), (+)(u+v) , (v+w), and (+)(u+w) are also independent of one another.

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