1.24 In the following questions we investigate geometric significance of spannin

Question

1.24 In the following questions we investigate geometric significance of spanning and linear independence. (a) Sketch the span of [1,2' in R (b) What do you guess that the span of ([1,2]'.[1, 1n R2 is? Draw a diagram to support your guess. Do you think that it is possible to find three linearly independent matrices in M(2, 1)'? On the basis of your answer to (c), do you think that it is possible to construct a 2 × 3 matrix with linearly independent columns? How about a 3 × 2 matrix with linearly independent rows? (c) (d) (e) Sketch (as best you can) the span of ([1,1,0F,[0,0, 1 in R (f) How does the span of [,1,,[1, 1,0]' in R3 compare with that in part (e)? Sketch the span of ill, 1,1], [2,2,2], in R3·Why is this picture so different from that in part (f)? Bring the phrase “ linearly dependent" into your discussion. (g)

Explanation / Answer

a)

span of [1,2] = c(1,2)

= (c,2c)

x= c, y = 2c

hence

y = 2x    {plot this line}

b)

span of [1,2] and [1,1] isR^2

this is because [1,2] and [1,1] are independent

[1,1] is y=x

c) No

d) no , it is not possible for linearly independent columns in 2*3 matrix

it is not possible for linearly independent rows   in 3*2 matrix

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