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Null space 3: Relation to linear independence

Understanding how the null space of a matrix relates to the linear independence of its column vectors. Created by Sal Khan.

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• Can someone explain me the physical meaning of null space? I understand it mathematically but don't get it in a practical sense. Thanks.
• A 'physical meaning' would come from the meaning of the numbers in the matrix. Look up using matrices to solve Kirchhoff’s Current Law for one example where the nullspace is used for an applied problem. Otherwise, if you just want some intuition: the nullspace for a matrix A is the set of vectors that are perpendicular to all the rows of A.
• Sal explains that the only way to the matrix vectors to be all linearly independent is if none of them is (may be represented as) a combination of the others. In which case the only solution is 0.

Then he says that for A.x = 0 to be true, x must be the zero vector. I guess I can make a conclusion by now, that the only way to satisfy the linear independency is if x = 0.

Is that wrong? Have I made any mistake in this thought?
• Rodrigo,

You haven't made a mistake, this is correct.

Basically Ax is the same as the ith column of A times the ith term in x (or each column of A times it's respective x term). If the columns of A are a linearly independent set, then the only way to multiply them all by some coefficients, and then add them all together and STILL get zero is if all of the coefficients are zero. Well in this case, the terms of x act like the coefficients of the columns of A. For the whole thing to be equal to the zero vector, all of the x terms must be 0. Or the "nullspace of A" is ONLY the zero vector.
• A Salman must have converted a matrix into a vector of vectors. This is confusing since I had always thought of matrices as composed of real numbers. In computer languages, such as Python there is a distinction between an "array" and a "list". A list can be a list of lists. But an array cannot be an array of arrays, well at least as far as I know. Can someone comment on what Salman did here?
• A matrix being a list of vectors comes straight out of the definition of a matrix. It is actually more correct to think of matrices as being composed of vectors rather than numbers.
• what does nullspace mean, physically and mathematically ?
• If `A x⃑ = 0⃑`
Then `N(A) = x⃑`
And vis-versa.

The nullspace of a matrix gives you a subspace.
`N(A) = x⃑ = V`
Every single value within that subspace will become the zero vector when transformed by A (which is what the original equation means).

If `N(A) ≠ 0⃑`
Then A isn't invertible.

Hope this gives you some understanding.
• So what is the actual purpose of null space? What does it signify?
• The nullspace is the set of all vectors v that, when multiplied by some matrix in the form Av, the result is the zero-vector. This is useful because it tells you every single vector that, when multiplied with A, will result in 0. That's why they call it the "null" space, or the space of vectors that will "nullify" or "zero out" a matrix. I've seen it used in low-level UX design (opengl) and in games where 3d vector manipulation is baked in (like Second Life). Also note the exciting idea that the null space represents the set of all vectors that are orthogonal to the matrix row vectors (dot product = 0).
• At Sal starts talking about Linearly Independence.
What if an N by N matrix wasn't Linearly Independent?
• Then the zero vector could be obtained by choosing an x not equal to zero in the equation Ax = 0. For instance, if in R^3, you had a 3x3 matrix A that could be multiplied by a vector x (where x isn't [0,0,0]) and the product was the zero vector ([0,0,0]), then the null space of A isn't trivial* which implies that the columns of A (i.e. the 3 R^3 vectors) are linearly dependent. This means that one of the vectors could be written as a combination of the other two.

In essence, if the null space is JUST the zero vector, the columns of the matrix are linearly independent. If the null space has more than the zero vector, the columns of the matrix are linearly dependent.

* trivial null space is just the zero vector.
• why does rref(A) have to be a SQUARE matrix?
• it is said in the previous comments above rref(A) does not have to be square matrix, it can also be in a form that square matrix and zeros below square form? I think also in the same way so rref(A) does not have to be a square matrix.
• Is an mxn matrix a tensor? A vector of vectors (m R^n row vectors or n R^m column vectors)?
• All mathematical objects in linear algebra are tensors. Matrices are order-2 tensors, vectors are order-1 tensors, and scalars are order-0 tensors.
• Can anyone explain in more intuitively way of null space is 0 when column vectors are linearly independent? Because other comment explained null space is a set of vector perpendicular to column vectors. I am confused about how 0 vectors perpendicular to linearly independent column vectors.
• The null space is not the set of vectors that are perpendicular to the column vectors. The nullspace is the set of vectors that perpendicular to the row vectors.

But you should think about it this way:
For a matrix A, the null space of A, denoted Null(A), is the set of all vectors x such that Ax=0, the right hand being the 0 vector.

Recall if the columns of A are linearly independent, then there is only the trivial solution to Ax=0, namely x=0.

This is why when the columns are linearly independent, the null space only has the 0 vector.