 # Linear Algebra Questions

## Transcribed Text

Question 1 Consider the matrix (a) Calculate the rret of the augmented matrix[A ]where denotesthe by identity matrix. 1.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 1.000 (b) invertible? yes . (c) your answer to (b) yes, write down the inverse of A. 1.500 3.500 9.00 -0.250 0.750 2.500 -1.250 2.750 7.500 (d) For each ofthe following statements decide if is true or false: (i) Ahas two pivot positions FALSE (ii) The equation Ax bas at leastone solution FAISE for each (iii) The columns f A are linearl independent TRA (iv) The lineartransformation x->Ax soneto Question The set of vectors 1) is not R² Prove this statement by (i) giving an example of vextor x: and scalar that xes, x- ax- (ii) giving example of vextorx and vector ysuchthatxes and yes, but x+yes. |x1-x2- by x+y. 1(x,+ y) -(x2++2))- Question Consider the subspaces S and S2 defined by the equations 5x1 10x-xx-0. (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 b1= b,= b.= (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) t Give geometric description of the set. The setisa

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