Algebra Determine whether or not each of the following sets of vectors in R4 is

Question

Algebra Determine whether or not each of the following sets of vectors in R4 is a basis for R4. In each case, write down a detailed reason for your Hint: For (a) and (b), you don't need to do any calculations! Calculus A drinking trough for cattle takes the form of a triangular prism, the cross-section of which is an isosceles triangle as sketched below. Its length is 3m, height 1.2m and width 0.8 m. The trough is filled with water at a rate of 5000 cm3 per minute. How fast is the height of the water in the trough rising when the water reaches 35 cm deep? 0.8m 3m 1.2m

Explanation / Answer

(a). The dimension of R4 is 4, whereas the set has 5 vectors. Hence, the given set cannot be a basis for R4.

(b). The vector (1,-2,3,0) = (1,0,0,0)-2(0,1,0,0)+3(0,0,1,0). Hence the given set cannot be a basis for R4.

( c). Let A be the 4x4 matrix with the vectors in the given set as columns. Then the RREF of A is

1

0

0

-2

0

1

0

1

0

0

1

3

0

0

0

0

It Implies that (2,-1,-1,-4) = -2(1,1,2,1) +(0,1,0,1) +3(0,0,1,-1). Hence the given set cannot be a basis for R4.

Please post the other question again separately.

1

0

0

-2

0

1

0

1

0

0

1

3

0

0

0

0

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