Consider the matrix
(a) Calculate the rret of the augmented matrix[A ]where denotesthe by identity matrix.
(c) your answer to (b) yes, write down the inverse of A.
(d) For each ofthe following statements decide if is true or false:
(i) Ahas two pivot positions
(ii) The equation Ax bas at leastone solution
(iii) The columns f A are linearl independent
(iv) The lineartransformation x->Ax
The set of vectors 1) is not R²
Prove this statement by
(i) giving an example of vextor x: and scalar that xes,
(ii) giving example of vextorx and vector ysuchthatxes and yes, but x+yes.
1(x,+ y) -(x2++2))-
Consider the subspaces S and S2 defined by the equations
(a) The vector (1,1, belongstr one of the subspaces Which oneisit?
(b) Determine basis fo the subspace you found in(a). Use as many fields as you need
(c) Write down the augmented matrix of the system of equations that needtobe solved to find the
coordinates of (1,1,-5) relativeto the basis. (use as man volumns as are needed)
Write down the coordinates of 1,1 ,-5)relative to the basis (Use as many rows as you need)
(d) Now consider the set of all vectors that belongto Determine vector equation that
describes this set (leave fields that are not needed free)
Give geometric description of the set.
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