When Af for each i, we shall have Af, because the operator Af will then be 0 on the range of Af.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

When the bridge was in use, boaters would signal the operator, who would start a diesel engine and rotate the bridge to the open position on a large gear.

When the important action shifts to a new position in the frame, the operator moves the scanner to follow it, creating the effect of a " pan " shot.

When a call is received, a jack lamp lights on the back panel and the operator responds by placing the rear cord into the corresponding jack and throwing the front key forward.

When the operator pulls down a cord, a pulley weight behind the switchboard pulls it down to prevent it from tangling.

When calling long distance, the calling party would give the name and city of the person desired, and the operator would advise the calling party to hang up and wait for the call to be completed.

When using the term tracking shot in this sense, the camera may be moved in ways not involving a camera dolly, such as via a Steadicam, via handheld camera operator, or by being panned on a tripod.

When they created the opening credits, Chris Carter found a video operator to stretch the face seen in the credits.

When used by a radio operator, a pseudonym is called a " handle ", especially in Citizens ' Band radio ; on the Appalachian Trail it is common to adopt or, more usually, be given by others a " trail name ".

When a fault or event occurs, a network component will often send a notification to the network operator using a protocol such as SNMP.

When using cable operator VoIP, a combined customer premises equipment device known as an embedded multimedia terminal adapter ( E-MTA ) will often be used.

When representing source code in Lisp, the first element of an S-expression is commonly an operator or function name and any remaining elements are treated as arguments.

:: When the Hamiltonian operator acts on the wavefunction Ψ, the result might be proportional to the same wavefunction Ψ.

When applied to a tangent vector X to the surface, the shape operator is the tangential component of the rate of change of the normal vector when moved along a curve on the surface tangent to X.

When the target was found the operator switches to " track " and uses a joystick to keep the target centred in the telescope.

When all of the children were old enough to attend school, Ann found employment as a linotype operator for the Bridgeport Post.

When postcodes are incomplete or missing altogether, the operator reads the post town name and inserts a code sufficient for outward sorting to the near-destination where others can further direct it.

When people wanted to talk to someone in the outside world, the would ask the operator for the number, and she'd connect them to the long-distance operator in Bowling Green, who would complete the call.

When an emergency happened in the pre-dial ( or " manual ") telephone era, the user simply picked up the telephone receiver and waited for the operator to answer " number, please?

When the long-term operator died unexpectedly, the carousel was put up for auction, with fears that it would leave Coney Island or be broken up for sale to collectors.

When synchronisation is lost and the phase break is deenergised, the train's operator should put the controller ( throttle ) into neutral and coast through an isolated phase break section.

When he turns briefly to literary style, in the Third Book, he again looks to the effect on the audience.

When one turns from the dombocs introduction to the laws themselves, it is difficult to uncover any logical arrangement.

When the radiometer is heated in the absence of a light source, it turns in the forward direction ( i. e. black sides trailing ).

When the total vapor pressure reaches the pressure surrounding the liquid, boiling occurs and liquid turns to gas throughout the bulk of the liquid.

When the caterpillar has grown enough, it turns into an immobile pupa.

When Hector turns to face his supposed brother to retrieve another spear he sees no one there.

When after several moments she still does not notice, he turns around and shoots himself.

When heat is applied to the meat a chemical reaction happens that turns the hemoglobin white.

When the king turns over temporalities to the clergy, he places them under his jurisdiction, from which later pronouncements of the popes cannot release them.

When Odras falls asleep, the Morrígan turns her into a pool of water.

When the signal light turns green, the race can begin.

When cycling was in its infancy, wooden indoor tracks were laid which resemble those of modern velodromes, consisting of two straights and slightly banked turns.

When the situation turns into a war between Japan and America, they establish contacts with the opposition in the Japanese government and are also used to eliminate a pair of Japanese AWACS planes.

When traveling through the turns at racing speed, which may exceed 85 km / h ( about 52 mph ), the banking attempts to match the natural lean of a bicycle moving through that curve.

When rapid parallel turns are quickly linked together, the upper body remains pointed down the fall line, and the lower body and skis are alternately driven into the snow in one direction and then the other.

When the incoming voltage falls below a predetermined level the SPS turns on its internal DC-AC inverter circuitry, which is powered from an internal storage battery.

When it finds itself pursued and in danger of capture, it throws itself from a precipice, and turns so aptly in falling, that it receives all the shock upon the horn, and so escapes safe and sound.

When one enters an elevator, it is expected that one turns around to face the doors.

When the turns are taken around the vertical pole they should be inside the previous turns.

When the driver turns to look back, the girl has vanished.

Hicks later agreed that the model missed important points of Keynesian theory, criticizing it as having very limited use beyond " a classroom gadget ", and criticizing equilibrium methods generally: " When one turns to questions of policy, looking towards the future instead of the past, the use of equilibrium methods is still more suspect.

When a drop of holy water falls on the talisman, the opal turns into a colorless stone and the Baroness dies soon thereafter.

" When Alienation turns Right ".

When she learns that he launched a full-scale bank heist instead, she injects him with concentrated U4 and turns him into a mutant, sending him after Blade.