1) Let V be the set of all ordered pairs (x, y), where x, y are...

  

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1) Let V be the set of all ordered pairs (x, y), where x, y are real numbers. Let a= (x, y) and b=(x be two elements in V. Define the addi tion as a +b = (x, y,)+(*2, y2 ) = (2x, - 3x,, y - Y2) the scalar multiplication as a (,, y1) = (a x, / 3,a y,/3). Show that Vis not a vector space. W of the properties are not satisfied? X1,. 2) Decide if the following statements are true or false. Provide arguments or a couterexample to suppi your answers: The set of matrices A with det(A)=D1 is a subspace in the vector space space Mx2(IR) of 2x2 mat (Ans. No because it does not contains the zero matrix). If S-span(u,,u2.,Un), then dim(S) = n. (Ans: True only if the vectors are linearly independent.) i. İi. %3D 3) Let V be the vector space of all polynomials of degree <3.Determine whether or not the set S= {r',? +1,r' +t+1 1} spans V? E)- (3) 3)(0 4) Show that the vectors and do not spans IR" 3 13 3 5) Let V be the vector space of all 2X2 real matrices. 0 1)(0 0) Show that the sets 1) Let V be the set of all ordered pairs (x, y), where x, y are real numbers. Let a= (x, y) and b=(x be two elements in V. Define the addi tion as a +b = (x, y,)+(*2, y2 ) = (2x, - 3x,, y - Y2) the scalar multiplication as a (,, y1) = (a x, / 3,a y,/3). Show that Vis not a vector space. W of the properties are not satisfied? X1,. 2) Decide if the following statements are true or false. Provide arguments or a couterexample to suppi your answers: The set of matrices A with det(A)=D1 is a subspace in the vector space space Mx2(IR) of 2x2 mat (Ans. No because it does not contains the zero matrix). If S-span(u,,u2.,Un), then dim(S) = n. (Ans: True only if the vectors are linearly independent.) i. İi. %3D 3) Let V be the vector space of all polynomials of degree <3.Determine whether or not the set S= {r',? +1,r' +t+1 1} spans V? E)- (3) 3)(0 4) Show that the vectors and do not spans IR" 3 13 3 5) Let V be the vector space of all 2X2 real matrices. 0 1)(0 0) Show that the sets