Find a basis for the given subspaces of R3 or R4 All vectors

Find a basis for the given subspaces of R^3 or R^4. All vectors of the matrix [a b c], where b = 0 All vectors of the form [a + c a - b b + c -a + b] All vectors of the form [a b c], where a - b + 5c=0

Solution

(a) Here we have to find the vectors that are linearly independent i.e the vectors that are not the scaler multiples of each other.

So following the restriction that b=0, and a and b are real, the two such vectors will be <1,0,0> and <0,0,1>it is because it is of form a(1,0,0)+c(0,0,1)

so two required vectors of the said form are (1,0,0) and (0,0,1)

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Part(b) :On using the same vectors we get the corresponding vector of (1,0,0) in given form as :

<a+c,a-b,b+c,-a+b>= < 1+0,1-0,0+0,-1+0> = <1,1,0,-1>

and corresponding vector of (0,0,1) will be <0+1,0-0,0+1,-0+1> = <1,0,1,1>

so two required vectors are <1,1,0,-1> and <1,0,1,1>

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PArt (c) , Again two vectors that make the basis ( one of its component can be written as a linear combination of other two components) and are in form a-b+5c =0 can be choosen as : <1,4,-1> and <3,-3,0>

Answer