a) Consider the following system of equations:
To consider a system of linear equations, we first represent it in matrix form as $AX = B$, where $A$ is the coefficient matrix, $X$ is the column vector ____ __________ _________ _______ _____ ________ ______ __________ _______.
_____ ____ ____ _________ ____.
______ _____ __________ _________ ______ ______ _______ _______ ________ _______.
_________ _________ __________ ___ ____ _______ ___ _______ _____ _____ _________ _____.
___ ___ _____ ___ _______ _______ _______.
_______ _______ __________ ___ ____ ______ ______.
__________ __________ _____ _______ ________ __________ _______ ________ __________ ________ ________.
________ ___ __________ ___ __________ ________ ____ _________ _______ ______ ___.
_______ ___ _____ ________ ___.
___ ____ _________ __________ _____ _____ __________ ________ ___ ______.
_______ ____ _______ __________ _____ _____ _____ ____.
_________ __________ ____ ______ ____.
_______ __________ ______ __________ ____ __________ _____.
__________ ____ _________ ____ _________ ________ __________ ________.
_______ __________ ______ _____ _________ ____ ___ ___ _________.
______ _________ ____ __________ ____ ________ _______.
_____ _________ _______ ________ ________ _________ ___ _______ _____.
_______ ___ __________ __________ __________ _______.
_______.
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a) Let
a) Which of the following are subspaces of R3 ? Justify your answer.
i) { }
ii) { }
For those subsets which are subspaces, find a basis.
b) Check that
Let and
Compute the scalar products
Find the vector equation of the plane determined by the points (1, 0,−1), (0, 1, 1) and (−1, 1, 0). Also find the point of intersection of the line
Which of the following are binary operations on ? Justify your answer.
i) The operation ▽ defined by
a) Check whether the matrices
Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) The Row Reduced Echelon form of any invertible square matrix is the identity matrix
dim(W1) dim(v)/2,dim(W2)
dim(v) /2 ,the W1 ∩ W2
{0}.
iii) If