Hi Sir thanks for these exquisite lectures. I have a question regarding null space. If we are given set of vectors from space of 2x2 Matrices how can we find null space of that, thabks.
The range has nothing to do with the bottom row. The range of a matrix is the column space of that matrix. To find the column space do row reduction. For any row with a leading 1 in reduced row echelon form look at that corresponding column of the original matrix and those columns will be a basis set for the range.
@@Jnglfvr Yes, and to see if a particular vector B is a column vector (belongs to the range), there must be only one solution that, Ax=B, or the system must be consistent.
You could call that the domain, but linear transformations/matrices are so well behaved that the domain is just always all of R^n itself, so it's not particularly useful to give it another name. The range, by contrast, might by R^m or a smaller smaller subspace inside of R^m.
"Input space": the set of vectors that can be put into the linear transformation = the set of vectors that you can multiply A by = R^n = domain of the linear transformation/matrix. "Output space": the set of vectors that can come out of the linear transformation = the set of vectors that you might get after multiplying A by some vector = the range of the linear transformation/matrix (which is a subspace of R^m).
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Hi Sir thanks for these exquisite lectures. I have a question regarding null space. If we are given set of vectors from space of 2x2 Matrices how can we find null space of that, thabks.
how to compute the range of a matrix if the bottom row is not all zero
The range has nothing to do with the bottom row. The range of a matrix is the column space of that matrix. To find the column space do row reduction. For any row with a leading 1 in reduced row echelon form look at that corresponding column of the original matrix and those columns will be a basis set for the range.
@@Jnglfvr Yes, and to see if a particular vector B is a column vector (belongs to the range), there must be only one solution that, Ax=B, or the system must be consistent.
Why isn't the set of all input vectors x that are elements R^n referred to as the domain?
You could call that the domain, but linear transformations/matrices are so well behaved that the domain is just always all of R^n itself, so it's not particularly useful to give it another name. The range, by contrast, might by R^m or a smaller smaller subspace inside of R^m.
what do u mean by input and output space at 4:18
"Input space": the set of vectors that can be put into the linear transformation = the set of vectors that you can multiply A by = R^n = domain of the linear transformation/matrix.
"Output space": the set of vectors that can come out of the linear transformation = the set of vectors that you might get after multiplying A by some vector = the range of the linear transformation/matrix (which is a subspace of R^m).
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