For each given set, addition operation, scalar multiplication determine whether
ID: 3031016 • Letter: F
Question
For each given set, addition operation, scalar multiplication determine whether the group is a vector space over the reals.Show work for all 10 properties.
The set of all 2x2 lower triangle matrices with normal addition and scalar multiplication. For each given set, addition operation, scalar multiplication determine whether the group is a vector space over the reals.
Show work for all 10 properties.
The set of all 2x2 lower triangle matrices with normal addition and scalar multiplication.
Show work for all 10 properties.
The set of all 2x2 lower triangle matrices with normal addition and scalar multiplication.
Explanation / Answer
A square matrix is called lower triangular if all the entries above the main diagonal are zero.
A vector space V is a set which is closed under finite vector addition and scalar multiplication. For a vector space, the scalars are members of a field F (say), and V is called a vector space over the field F.
If V is to be a vector space, then V must satisfy the following ten conditions for all elements (say) X, Y, Z V and any scalars p,q F:
Let us now check whether the set M of all 2x2 lower triangular matrices satisfies the above 10 conditions. Let X, Y, Z M and let the arbitrary scalars p, q F, where F is a field over which M is defined. Then
Thus M, the set of all 2x2 lower triangular matrices satisfies the above 10 conditions and hence M is a vector space.
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