Suppose that 𝐯1, …, 𝐯n form a basis for ℝ^n. Let A be a nonsingular matrix. Prove that A 𝐯1, …,

Suppose that 𝐯1, …, 𝐯n form a basis for ℝ^n. Let A be a...

Key Concepts

Basis

A basis of a vector space is a set of vectors that are both linearly independent and span the entire space. This means every vector in the space can be uniquely expressed as a linear combination of the basis vectors. In R?, any set of n independent vectors that span ?? forms a basis.

Invertible (Nonsingular) Matrix

An invertible or nonsingular matrix is a square matrix that has an inverse. This property guarantees that the linear transformation associated with the matrix is bijective (both injective and surjective), meaning it does not collapse the space and each output is uniquely mapped from an input.

Standard Basis and Its Transformation

The standard basis for ?? consists of vectors with a 1 in one coordinate and 0 in all others. When an invertible matrix acts on the standard basis, each transformed vector is the corresponding column of the matrix. These column vectors are guaranteed to be linearly independent and span ??, thus forming a new basis for the space.

Linear Transformation Mapping Basis to Basis

A fundamental property of invertible linear transformations is that they map bases to bases. When the transformation defined by an invertible matrix is applied to a basis of the vector space, the resulting set of vectors remains linearly independent and continues to span the space, thereby forming a new basis.

Transcript

00:01 Hi there.

00:02 For the following exercise we need to prove we need to work with a proof.

00:06 Okay, so first we're going to consider an inner transformation ta defined with an invertible matrix and further we're going to consider that you want you to until you and form a basis for the euclidean space and dimensional.

00:28 Okay, so we need to show we need to prove actually that the set of transformed vectors u u uh with respect to the this transformation t is also a basis for rn okay so let's start by the the fact that this set form a basis for r n okay so if we know that you u1, u2 until un form a basis for rn, okay? this is equivalent to say that this set u1 until un is linearly independent and it span rn.

01:40 Okay that's the conditions to be a basis so on one hand the linearly independence can be represented by the sum from i equals to one up until n of c i ui equals to zero if and only if all these cis are equals to zero with i equals to one until n okay this c i is are just constants okay but these the linear independence means that these linear combinations of the basis elements are equals to zero if and only if all these constants are equals to zero and on the other hand we need to show that the span is that for a generic vector v in our n then there exists well then this vector v can be represented uniquely from c i equals to 1 until n of c i as a linear combination of the basis elements ui okay so there is of course there is a unique unique set of c i's that satisfy this equation okay so we can represent any vector as a linear combination of this basis elements okay so now you can observe that for the linearly independent condition let me put it in a different color the linearly independent condition if we apply the transformation t a we have here the linear transformation applied to the sum from i equals to one up until n of c i u i equals to the linear transformation applied to the here is the vector zero i forgot to mention here we have the zero vector the zero vector so we have this we know and this is inherited this equation is inherited from the previous exercise so basically this is saying that this is equals to zero if all the cis are equals to zero okay for all the i's okay we just apply this transformation to this this this equation here okay so we have that this is equals to this we apply just the linear transformation to the previous equation and we know that the a linear transformation map the zero vector to the zero vector.

05:02 In this case we're map into the same space so we obtain the same zero vector and then we by linearity property of the operation we have that this is equals to the sum from i equals to one until n of c i t a ui equals to zero...

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