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Let V be the vector space of all 2 x 2 matrices over the field R of real numbers and 
is a linear transformation defined by T(A) = AB - BA, then what is the dimension of the kernel of T?
is a linear transformation defined by T(A) = AB - BA, then what is the dimension of the kernel of T?Correct answer is '2'. Can you explain this answer?
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Verified Answer
Let V be the vector space of all 2 x 2 matrices over the field R of re...
Let U (F) and V(F) be two vector spaces and let T be a linear transformation from U to V, then the null space of T written as N(T) is the set of all vectors a in U such that T(α) = Of zero vector of V)
The null space of T is also called the kernel of T.
Now the standard basis for a vector space of all 2x2 matrices over field R of real number is {e1, e2, e3, e4}
where
we have, T(A) = AB - BA, where B =
so,
Thus. T (e1), T(e2), T(e3) and T(e4) span the range of T
Null space
let
∈ null space of T
This system of equations have two independent variables hence, it have two independent solutions.
⇒ The dimension of null space is 2.
The null space of T is also called the kernel of T.
Now the standard basis for a vector space of all 2x2 matrices over field R of real number is {e1, e2, e3, e4}
where
we have, T(A) = AB - BA, where B =
so,
Thus. T (e1), T(e2), T(e3) and T(e4) span the range of T
Null space
let
∈ null space of TThis system of equations have two independent variables hence, it have two independent solutions.
⇒ The dimension of null space is 2.
Most Upvoted Answer
Let V be the vector space of all 2 x 2 matrices over the field R of re...
Let U (F) and V(F) be two vector spaces and let T be a linear transformation from U to V, then the null space of T written as N(T) is the set of all vectors a in U such that T(α) = Of zero vector of V)
The null space of T is also called the kernel of T.
Now the standard basis for a vector space of all 2x2 matrices over field R of real number is {e1, e2, e3, e4}
where
we have, T(A) = AB - BA, where B =
so,
Thus. T (e1), T(e2), T(e3) and T(e4) span the range of T
Null space
let ∈ null space of T
This system of equations have two independent variables hence, it have two independent solutions.
⇒ The dimension of null space is 2.
The null space of T is also called the kernel of T.
Now the standard basis for a vector space of all 2x2 matrices over field R of real number is {e1, e2, e3, e4}
where
we have, T(A) = AB - BA, where B =
so,
Thus. T (e1), T(e2), T(e3) and T(e4) span the range of T
Null space
let
∈ null space of TThis system of equations have two independent variables hence, it have two independent solutions.
⇒ The dimension of null space is 2.
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Community Answer
Let V be the vector space of all 2 x 2 matrices over the field R of re...
Let U (F) and V(F) be two vector spaces and let T be a linear transformation from U to V, then the null space of T written as N(T) is the set of all vectors a in U such that T(α) = Of zero vector of V)
The null space of T is also called the kernel of T.
Now the standard basis for a vector space of all 2x2 matrices over field R of real number is {e1, e2, e3, e4}
where
we have, T(A) = AB - BA, where B =
so,
Thus. T (e1), T(e2), T(e3) and T(e4) span the range of T
Null space
let ∈ null space of T
This system of equations have two independent variables hence, it have two independent solutions.
⇒ The dimension of null space is 2.
The null space of T is also called the kernel of T.
Now the standard basis for a vector space of all 2x2 matrices over field R of real number is {e1, e2, e3, e4}
where
we have, T(A) = AB - BA, where B =
so,
Thus. T (e1), T(e2), T(e3) and T(e4) span the range of T
Null space
let
∈ null space of TThis system of equations have two independent variables hence, it have two independent solutions.
⇒ The dimension of null space is 2.
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Let V be the vector space of all 2 x 2 matrices over the field R of real numbers andis a linear transformation defined by T(A) = AB - BA, then what is the dimension of the kernel of T?Correct answer is '2'. Can you explain this answer?
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